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Trigonometry Honors -Phase 2

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PHASE 2 INSTRUCTIONAL CONTINUITY PLAN

BEGINNING APRIL 16, 2020

2

Topic Number Topic Name Date

Completed

Section 13-1

First-Degree Trigonometric Equations Lesson: Read pages 518 – 524 paying attention to the “Examples.”

Practice: Complete exercises #2, #3- 25 odd on pages 524-525

Check your answers in the back of this attachment.

Section 13-2

Using Factoring to Solve Trigonometric Equations Lesson: Read pages 526 – 529 paying attention to the “Examples.”

Practice: Complete exercises #1- 21 odd on pages 529-530


Section 13-3

Using the Quadratic Formula to Solve Trigonometric Equations Lesson: Read pages 530 – 533 paying attention to the “Examples.”

Practice: Complete exercises #1- 13 odd on page 534


3

Section 13-4

Using Substitution to Solve Trigonometric Equations Involving More than One Function Lesson: Read pages 534 – 537 paying attention to the “Examples.” Practice: Complete exercises #1, #3 – 9, #14 on page 537 Check your answers in the back of this attachment.

Section 13-5

Using Substitution to Solve Trigonometric Equations Lesson: Read pages 538 – 540 paying attention to the “Examples.” Practice: Complete exercises #1- 13 odd on pages 540 - 541 Check your answers in the back of this attachment.

Chapter Summary

Chapter Summary Lesson: Read page 542 Practice: Complete exercises #1- 6, 11-17 odd on page 543 Check your answers in the back of this attachment.

Section 8.8

Vectors Lesson: Read pages 729 – 742 paying close attention to the “Examples” and completing the “Try It” questions. The “Try It” questions can be check at the end of this attachment. Practice: Complete exercises #1- 47 odd, #61 on pages 743 -745 Check your answers in the back of this attachment.

CHAPTER

13

518

CHAPTERTABLE OF CONTENTS

13-1 First-Degree TrigonometricEquations

13-2 Using Factoring to SolveTrigonometric Equations

13-3 Using the Quadratic Formulato Solve TrigonometricEquations

13-4 Using Substitution to SolveTrigonometric EquationsInvolving More Than OneFunction

13-5 Using Substitution to SolveTrigonometric EquationsInvolving Different AngleMeasures

Chapter Summary

Vocabulary

Review Exercises

Cumulative Review

TRIGONOMETRICEQUATIONS

The triangle is a rigid figure, that is, its shape cannotbe changed without changing the lengths of its sides.This fact makes the triangle a basic shape in construc-tion. The theorems of geometry give us relationshipsamong the measures of the sides and angles of triangle.Precisely calibrated instruments enable surveyors toobtain needed measurements. The identities and for-mulas of trigonometry enable architects and builders toformulate plans needed to construct the roads, bridges,and buildings that are an essential part of modern life.

14411C13.pgs 8/12/08 1:56 PM Page 518

A trigonometric equation is an equation whose variable is expressed in terms ofa trigonometric function value. To solve a trigonometric equation, we use thesame procedures that we used to solve algebraic equations. For example, in theequation 4 sin u 1 5 5 7, sin u is multiplied by 4 and then 5 is added. Thus, tosolve for sin u, first add the opposite of 5 and then divide by 4.

4 sin u 1 5 5 7 2 5 5 25

4 sin u 5 2

5

sin u 5

We know that sin 30° 5 , so one value of u is 30°; u1 5 30. We also knowthat since sin u is positive in the second quadrant, there is a second-quadrant angle, u2, whose sine is . Recall the relationship between an angle inany quadrant to the acute angle called the reference angle. The following tablecompares the degree measures of u from 290° to 360°, the radian measures ofu from 2 to 2p, and the measure of its reference angle.

The reference angle for the second-quadrant angle whose sine is has a

degree measure of 30° and u2 5 180° 2 30° or 150°. Therefore, sin u2 5 . For 0° � u , 360°, the solution set of 4 sin u 1 5 5 7 is {30°, 150°}. In radian mea-sure, the solution set is .

In the example given above, it was possible to give the exact value of u thatmakes the equation true. Often it is necessary to use a calculator to find anapproximate value. Consider the solution of the following equation.

5 cos u 1 7 5 3

5 cos u 5 24

cos u 5 2

u 5 arccos A245 B

45

Up6 , 5p

6 V

12

12

p2

12

12

12

24

4 sin u4

13-1 FIRST-DEGREE TRIGONOMETRIC EQUATIONS

First-Degree Trigonometric Equations 519

Fourth First Second Third FourthQuadrant Quadrant Quadrant Quadrant Quadrant

Angle 290° , u , 0° 0° , u , 90° 90° , u , 180° 180° , u , 270° 270° , u , 360°, u , 0 , u , p p , u , , u , 2p

Reference 2u u 180° 2 u u 2 180° 360° 2 u

Angle 2u u p 2 u u 2 p 2p 2 u

3p2

3p2

p20 , u , p

22p2

14411C13.pgs 8/12/08 1:56 PM 19

When we use a calculator to find u, the calculator will return the value ofthe function y 5 arccos x whose domain is 0° � x � 180° in degree measure or0 � x � p in radian measure.

In degree measure:

ENTER: 4 DISPLAY:

5

To the nearest degree, one value of u is143°. In addition to this second-quadrantangle, there is a third-quadrant angle suchthat cos u 5 . To find this third-quadrantangle, find the reference angle for u.

Let R be the measure of the referenceangle of the second-quadrant angle. That is, R is the acute angle such that cos u 5 2cos R.

R 5 180° 2 u 5 180° 2 143° 5 37°

The measure of the third-quadrant angle is:

u 5 R 1 180°

u 5 37° 1 180°

u 5 217°

For 0° � u � 360°, the solution set of 5 cos u 1 7 5 3 is {143°, 217°}.If the value of u can be any angle measure, then for all integral values of n,u 5 143 1 360n or u 5 217 1 360n.

245

ENTER) 1 4 3 . 1 3 0 1 0 2 4c o s - 1 ( - 4 / 5 )

�(-)COS�12nd

520 Trigonometric Equations

Procedure

To solve a linear trigonometric equation:

1. Solve the equation for the function value of the variable.

2. Use a calculator or your knowledge of the exact function values to writeone value of the variable to an acceptable degree of accuracy.

3. If the measure of the angle found in step 2 is not that of a quadrantal angle,find the measure of its reference angle.

4. Use the measure of the reference angle to find the degree measures of eachsolution in the interval 0° � u , 360° or the radian measures of each solu-tion in the interval 0 � u , 2p.

5. Add 360n (n an integer) to the solutions in degrees found in steps 2 and 4 towrite all possible solutions in degrees.Add 2pn (n an integer) to the solutionsin radians found in steps 2 and 4 to write all possible solutions in radians.

y

x37°

143°

14411C13.pgs 8/12/08 1:56 PM Page 520

The following table will help you find the locations of the angles that satisfytrigonometric equations. The values in the table follow from the definitions ofthe trigonometric functions on the unit circle.

EXAMPLE 1

Find the solution set of the equation 7 tan u 5 1 tan u in the interval 0° � u , 360°.

Solution How to Proceed

(1) Solve the equation for tan u: 7 tan u 5 1 tan u

6 tan u 5

tan u 5

(2) Since tan u is positive, u1 can be a u1 5 30°first-quadrant angle:

(3) Since u is a first-quadrant angle, R 5 30°R 5 u:

(4) Tangent is also positive in the third u2 5 180° 1 Rquadrant. Therefore, there is a u2 5 180° 1 30° 5 210°third-quadrant angle such that

tan u 5 . In the third quadrant,

u2 5 180° 1 R:

Answer The solution set is {30°, 210°}.

!33

!33

2!3

2!3

2!3


Sign of a and b (0 * |a| * 1, b � 0)

1 2

sin u 5 a Quadrants I and II Quadrants III and IV

cos u 5 a Quadrants I and IV Quadrants II and III

tan u 5 b Quadrants I and III Quadrants II and IV

14411C13.pgs 8/12/08 1:56 PM Page 521

EXAMPLE 2

Find, to the nearest hundredth, all possible solutions of the following equationin radians:

3(sin A 1 2) 5 3 2 sin A


(1) Solve the equation for sin A: 3(sin A 1 2) 5 3 2 sin A

3 sin A 1 6 5 3 2 sin A

4 sin A 5 23

sin A 5

(2) Use a calculator to find one value of A (be sure that the calculator is in RADIAN mode):

(3) Find the reference angle: R 5 2A 5 2(20.848) 5 0.848

(4) Sine is negative in quadrants In quadrant III: A1 5 p 1 0.848 � 3.99III and IV. Use the In quadrant IV: A2 5 2p 2 0.848 � 5.44reference angle to find a value of A in each of these quadrants:

(5) Write the solution set: {3.99 1 2pn, 5.44 1 2pn} Answer

Note: When sin21 was entered, the calculator returned the value in the

interval , the range of the inverse of the sine function. This is the measure of a fourth-quadrant angle that is a solution of the equation. How-ever, solutions are usually given as angle measures in radians between 0 and 2p plus multiples of 2p. Note that the value returned by the calculator is 5.43 1 2p (21) � 20.85.

EXAMPLE 3

Find all possible solutions to the following equation in degrees:

12 ( sec u 1 3) 5 sec u 1 5

2

2p2 # u # p

2

A234 B

234


ENTER: 3 4

DISPLAY:

One value of A is 20.848.

- . 8 4 8 0 6 2 0 7 9s i n - 1 ( - 3 / 4 )

ENTER

)�(-)SIN�12nd

14411C13.pgs 8/12/08 1:56 PM Page 522

Solution (1) Solve the equation for sec u:

(2) Rewrite the equation in terms of cos u:

(3) Use a calculator to find one value of u:

(4) Find the reference angle: R 5 180 2 120 5 60°

(5) Cosine is also negative in In quadrant III: u2 5 180 1 60 5 240°quadrant III. Use the reference angle to find a value of u2 in quadrant III:

(6) Write the solution set: {120 1 360n, 240 1 360n} Answer

Trigonometric Equations and the GraphingCalculator

Just as we used the graphing calculator to approximate the irrational solutionsof quadratic-linear systems in Chapter 5, we can use the graphing calculator toapproximate the irrational solutions of trigonometric equations. For instance,

3(sin A 1 2) 5 3 2 sin A

from Example 2 can be solved using the intersect feature of the calculator.STEP 1. Treat each side of the equation as a function.

Enter Y1 5 3(sin X 1 2) and Y2 5 3 2 sin X into the menu.

STEP 2. Using the following viewing window:

Xmin 5 0, Xmax 5 2p, Xscl 5 ,Ymin 5 0, Ymax 5 10

and with the calculator set to radian

mode, the functions.GRAPH

p6

Y�

cos u 5 212

22 5 sec u

sec u 1 3 5 2 sec u 1 5

12 ( sec u 1 3) 5 sec u 1 5

2


ENTER: 21 2

DISPLAY:

Cosine is negative in quadrant II so u1 5 120°.

1 2 0c o s - 1 ( - 1 / 2 )

ENTER

)�COS�12nd

14411C13.pgs 8/12/08 1:56 PM Page 523

STEP 3. The solutions are the x-coordinates of the intersection points of the graphs. We can find the intersection points by using the intersect function. Press 5 to select both curves.When the calculator asks you for a guess, move the cursor near one of the intersection points using the arrow keys and then press .Repeat this process to find the other intersection point.

As before, the solutions in the interval 0 � u , 2p are approximately 3.99 and5.44. The solution set is {3.99 1 2pn, 5.44 1 2pn}.

Writing About Mathematics

1. Explain why the solution set of the equation 2x 1 4 5 8 is {2} but the solution set of theequation 2 sin x 1 4 5 8 is { }, the empty set.

2. Explain why 2x 1 4 5 8 has only one solution in the set of real numbers but the equation 2 tan x 1 4 5 8 has infinitely many solutions in the set of real numbers.

Developing SkillsIn 3–8, find the exact solution set of each equation if 0° � u , 360°.

3. 2 cos u 2 1 5 0 4. 3 tan u 1 5 0

5. 4 sin u 2 1 5 2 sin u 1 1 6. 5(cos u 1 1) 5 5

7. 3(tan u 2 2) 5 2 tan u 2 7 8. sec u 1 5

In 9–14, find the exact values for u in the interval 0 � u , 2p.

9. 3 sin u 2 5 sin u 10. 5 cos u 1 3 5 3 cos u 1 5

11. tan u 1 12 5 2 tan u 1 11 12. sin u 1 5

13. 3 csc u 1 5 5 csc u 1 9 14. 4(cot u 1 1) 5 2(cot u 1 2)

!22!2

!3

2!2!2

!3

Exercises

I n t e r s e c t i o nX = 5 . 4 3 5 1 2 3 2 Y = 3 . 7 5

*I n t e r s e c t i o nX = 3 . 9 8 9 6 5 4 7 Y = 3 . 7 5

*

ENTER

ENTERENTERCALC2nd


14411C13.pgs 8/12/08 1:56 PM Page 524

In 15–20, find, to the nearest degree, the measure of an acute angle for which the given equation istrue.

15. sin u 1 3 5 5 sin u 16. 3 tan u 2 1 5 tan u 1 9

17. 5 cos u 1 1 5 8 cos u 18. 4(sin u 1 1) 5 6 2 sin u

19. csc u 2 1 5 3 csc u 2 11 20. cot u 1 8 5 3 cot u 1 2

In 21–24, find, to the nearest tenth, the degree measures of all u in the interval 0° � u , 360° thatmake the equation true.

21. 8 cos u 5 3 2 4 cos u 22. 5 sin u 2 1 5 1 2 2 sin u

23. tan u 2 4 5 3 tan u 1 4 24. 2 2 sec u 5 5 1 sec u

In 25–28, find, to the nearest hundredth, the radian measures of all u in the interval 0 � u , 2p thatmake the equation true.

25. 10 sin u 1 1 5 3 2 2 sin u 26. 9 2 2 cos u 5 8 2 4 cos u

27. 15 tan u 2 7 5 5 tan u 2 3 28. cot u 2 6 5 2 cot u 1 2

Applying Skills

29. The voltage E (in volts) in an electrical circuit is given by the function

E 5 20 cos (pt)where t is time in seconds.

a. Graph the voltage E in the interval 0 � t � 2.

b. What is the voltage of the electrical circuit when t 5 1?

c. How many times does the voltage equal 12 volts in the first two seconds?

d. Find, to the nearest hundredth of a second, the times in the first two seconds when thevoltage is equal to 12 volts.

(1) Let u 5 pt. Solve the equation 20 cos u 5 12 in the interval 0 � u , 2p.

(2) Use the formula u 5 pt and your answers to part (1) to find t when 0 � u , 2p andthe voltage is equal to 12 volts.

30. A water balloon leaves the aircannon at an angle of u with theground and an initial velocity of40 feet per second. The water bal-loon lands 30 feet from the can-non. The distance d traveled bythe water balloon is given by theformula

where v is the initial velocity.

d 5 132 v2

sin 2u


30 ft

θ

14411C13.pgs 8/12/08 1:56 PM Page 525

a. Let x 5 2u. Solve the equation 30 5 to the nearest tenth of a degree.

b. Use the formula x 5 2u and your answer to part a to find the measure of the anglethat the cannon makes with the ground.

31. It is important to understand the underlying mathematics before using the calculator tosolve trigonometric equations. For example, Adrian tried to use the intersect feature of his graphing calculator to find the solutions of the equation cot u 5 sin in the interval 0 � u � p but got an error message. Follow the steps that Adrian used to solve the equation:

(1) Enter Y1 5 and Y2 5 sin into the menu.

(2) Use the following viewing window to graph the equations:

Xmin 5 0, Xmax 5 p, Xscl 5 , Ymin 5 25, Ymax 5 5

(3) The curves seem to intersect at . Press 5

to select both curves. When the calculator asks for a guess, move the cursor

near the intersection point using the arrow keys and then press .

a. Why does the calculator return an error message?

b. Is u 5 a solution to the equation? Explain.

We know that the equation 3x2 2 5x 2 2 5 0 can be solved by factoring the leftside and setting each factor equal to 0. The equation 3 tan2 u 2 5 tan u 2 2 5 0can be solved for tan u in a similar way.

3x2 2 5x 2 2 5 0 3 tan2 u 2 5 tan u 2 2 5 0

(3x 1 1)(x 2 2) 5 0 (3 tan u 1 1)(tan u 2 2) 5 0

3x 1 1 5 0 x 2 2 5 0 3 tan u 1 1 5 0 tan u 2 2 5 0

3x 5 21 x 5 2 3 tan u 5 21 tan u 5 2

x 5 2 tan u 5

In the solution of the algebraic equation, the solution is complete. The solu-tion set is . In the solution of the trigonometric equation, we must nowfind the values of u.

There are two values of u in the interval 0° � u , 360° for which tan u 5 2,one in the first quadrant and one in the third quadrant. The calculator will dis-play the measure of the first-quadrant angle, which is also the reference anglefor the third-quadrant angle.

U213, 2V

213

13

13-2 USING FACTORING TO SOLVE TRIGONOMETRIC EQUATIONS

p2

ENTER

ENTERENTERCALC2ndAp2 , 0 B

p6

Y�AX 2 p2 B1

tan X

A u 2 p2 B

132(40)2

sin x


14411C13.pgs 8/12/08 1:56 PM Page 526

ENTER: 2 DISPLAY:

To the nearest tenth of a degree, the measure of u in the first quadrant is63.4. This is also the reference angle for the third-quadrant angle.

In quadrant I: u1 5 63.4°In quadrant III: u2 5 180 1 R 5 180 1 63.4 5 243.4°

There are two values of u in the interval 0° � u , 360° for which tan u 5 ,one in the second quadrant and one in the fourth quadrant.

ENTER: 21 3 DISPLAY:

The calculator will display the measure of a fourth-quadrant angle, which isnegative. To the nearest tenth of a degree, one measure of u in the fourth quad-rant is 218.4. The opposite of this measure, 18.4, is the measure of the referenceangle for the second- and fourth-quadrant angles.

In quadrant II: u3 5 180 2 18.4 5 161.6°In quadrant IV: u4 5 360 2 18.4 5 341.6°

The solution set of 3 tan2 u 2 5 tan u 2 2 5 0 is

{63.4°, 161.6°, 243.4°, 341.6°}

when 0° � u , 360°.

EXAMPLE 1

Find all values of u in the interval 0 � u , 2p for which 2 sin u 2 1 5 .

Solution How to Proceed(1) Multiply both sides of the

equation by sin u:

(2) Write an equivalent equation with 0 as the right side:

(3) Factor the left side:

(4) Set each factor equal to 0 and solve for sin u:

(5) Find all possible values of u:

Answer u 5 3p2

3sin u

ENTER - 1 8 . 4 3 4 9 4 8 8 2t a n - 1 ( - 1 / 3 )

)�TAN�12nd

213

6 3 . 4 3 4 9 4 8 8 2t a n - 1 ( 2 )

ENTER)TAN�12nd

Using Factoring to Solve Trigonometric Equations 527

2 sin u 2 1 5

2 sin2 u 2 sin u 5 32 sin2 u 2 sin u 2 3 5 0

(2 sin u 2 3)(sin u 1 1) 5 0

2 sin u 2 3 5 0 sin u 1 1 5 0

2 sin u 5 3 sin u 5 21

sin u 5

There is no value of u such that sin u . 1. For sin u 5 21, u 5 .3p

2

32

3sin u

14411C13.pgs 8/12/08 1:56 PM Page 527

EXAMPLE 2

Find the solution set of 4 sin2 A 2 1 5 0 for the degree measures of A in theinterval 0° � A , 360°.

Solution METHOD 1 METHOD 2

Factor the left side.

4 sin2 A 2 1 5 0

(2 sin A 2 1)(2 sin A 1 1) 5 0

2 sin A 2 1 5 0 2 sin A 1 1 5 0

2 sin A 5 1 2 sin A 5 21

sin A 5 sin A 5

If sin A 5 , A 5 30° or A 5 150°. If sin A 5 , A 5 210° or A 5 330°.

Answer {30°, 150°, 210°, 330°}

Note: The graphing calculator does not use the notation sin2 A, so we must enterthe square of the trig function as (sin A)2 or enter sin (A)2. For example, tocheck the solution A 5 30° for Example 2:

ENTER: 4 30 1

4 30 1

DISPLAY:

Factoring Equations with Two TrigonometricFunctions

To solve an equation such as 2 sin u cos u 1 sin u 5 0, it is convenient to rewritethe left side so that we can solve the equation with just one trigonometric func-tion value. In this equation, we can rewrite the left side as the product of twofactors. Each factor contains one function.

04 ( s i n ( 3 0 ) ) 2 - 1

04 s i n ( 3 0 ) 2 - 1

ENTER�x2)SIN

ENTER�x2))SIN(

212

12

212

12


Solve for sin2 A and take the squareroot of each side of the equation.

4 sin2 A 2 1 5 0

4 sin2 A 5 1

sin2 A 5

sin A 5 612

14

14411C13.pgs 8/12/08 1:56 PM Page 528

2 sin u cos u 1 sin u 5 0

sin u (2 cos u 1 1) 5 0

sin u 5 0 2 cos u 1 1 5 0

u1 5 0° 2 cos u 5 21

u2 5 180° cos u 5

Since cos 60° 5 , R 5 60°.

In quadrant II: u3 5 180 2 60 5 120°In quadrant III: u4 5 180 1 60 5 240°

The solution set is {0°, 120°, 180°, 240°}.

EXAMPLE 3

Find, in radians, all values of u in the interval 0 � u � 2p that are in the solutionset of:

sec u csc u 1 5 0

Solution Factor the left side of the equation and set each factor equal to 0.

sec u csc u 1 csc u 5 0

csc u 5 0

csc u 5 0 ✘ sec u 1 5 0

No solution sec u 5 2

cos u 5

Since cos , R 5 .

In quadrant II: u 5 p 2 5

In quadrant III: u 5 p 1 5

Answer


1. Can the equation tan u 1 sin u tan u 5 1 be solved by factoring the left side of the equa-tion? Explain why or why not.

2. Can the equation 2(sin u)(cos u) 1 sin u 1 2 cos u 1 1 5 0 be solved by factoring the leftside of the equation? Explain why or why not.

Exercises

U3p4 , 5p

4 V

5p4

p4

3p4

p4

p4

p4 5

!22

2 1!2 5 2!22

!2

!2

Asec u 1 !2B!2

!2 csc u

12

212

Using Factoring to Solve Trigonometric Equations 529

14411C13.pgs 8/12/08 1:56 PM Page 529

Developing SkillsIn 3–8, find the exact solution set of each equation if 0° � u , 360°.

3. 2 sin2 u 1 sin u 2 1 5 0 4. 3 tan2 u 5 1

5. tan2 u 2 3 5 0 6. 2 sin2 u 2 1 5 0

7. 6 cos2 u 1 5 cos u 2 4 5 0 8. 2 sin u cos u 1 cos u 5 0

In 9–14, find, to the nearest tenth of a degree, the values of u in the interval 0° � u , 360° that sat-isfy each equation.

9. tan2 u 2 3 tan u 1 2 5 0 10. 3 cos2 u 2 4 cos u 1 1 5 0

11. 9 sin2 u 2 9 sin u 1 2 5 0 12. 25 cos2 u 2 4 5 0

13. tan2 u 1 4 tan u 2 12 5 0 14. sec2 u 2 7 sec u 1 12 5 0

In 15–20, find, to the nearest hundredth of a radian, the values of u in the interval 0 � u , 2p thatsatisfy the equation.

15. tan2 u 2 5 tan u 1 6 5 0 16. 4 cos2 u 2 3 cos u 5 1

17. 5 sin2 u 1 2 sin u 5 0 18. 3 sin2 u 1 7 sin u 1 2 5 0

19. csc2 u 2 6 csc u 1 8 5 0 20. 2 cot2 u 2 13 cot u 1 6 5 0

21. Find the smallest positive value of u such that 4 sin2 u 2 1 5 0.

22. Find, to the nearest hundredth of a radian, the value of u such that sec u 5 and , u , p.

23. Find two values of A such that (sin A)(csc A) 5 2sin A.

Not all quadratic equations can be solved by factoring. It is often useful or nec-essary to use the quadratic formula to solve a second-degree trigonometricequation.

The trigonometric equation 2 cos2 u 2 4 cos u 1 1 5 0 is similar in form tothe algebraic equation 2x2 2 4x 1 1 5 0. Both are quadratic equations that can-not be solved by factoring over the set of integers but can be solved by using thequadratic formula with a 5 2, b 5 24, and c 5 1.

13-3 USING THE QUADRATIC FORMULA TO SOLVE TRIGONOMETRIC EQUATIONS

p2

5sec u


14411C13.pgs 8/12/08 1:56 PM Page 530

Algebraic equation: Trigonometric equation:

2x2 2 4x 1 1 5 0 2 cos2 u 2 4 cos u + 1 5 0

x 5 cos u 5

x 5 cos u 5

x 5 cos u 5

x 5 cos u 5

x 5 cos u 5

There are no differences between the two solutions up to this point.

However, for the algebraic equation, the solution is complete.There are two val-

ues of x that make the equation true: x 5 or x 5 .For the trigonometric equation, there appear to be two values of cos u. Can

we find values of u for each of these two values of cos u?

cos u 5 �

There is no value of u such that cos u . 1.

cos u 5 �

There are values of u in the first quadrant and in the fourth quadrant suchthat cos u is a positive number less than 1. Use a calculator to approximate thesevalues to the nearest degree.

ENTER: 2 DISPLAY:

2

2

To the nearest degree, the value of u in the first quadrant is 73°. This is also the value of the reference angle. Therefore, in the fourth quadrant,u 5 360° 2 73° or 287°.

In the interval 0° � u , 360°, the solution set of 2 cos2 u 2 4 cos u 1 1 5 0 is:

{73°, 287°}

ENTER)�

))�¯2nd7 2 . 9 6 8 7 5 1 5 4

c o s - 1 ( ( 2 - √ ( 2 ) ) / 2 )�(COS�12nd

2 2 1.4142 5 0.2932 2 "2

2CASE 2

1 1 1.4142 5 1.2072 1 !2

2CASE 1

2 2 !22

2 1 !22

2 6 !22

2 6 !22

4 6 !84

4 6 !84

4 6 !16 2 84

4 6 !16 2 84

2(24) 6 "(24)2 2 4(2)(1)2(2)

2(24) 6 "(24)2 2 4(2)(1)2(2)

2b 6 "b2 2 4ac2a

2b 6 "b2 2 4ac2a

Using the Quadratic Formula to Solve Trigonometric Equations 531

14411C13.pgs 8/12/08 1:56 PM Page 531

EXAMPLE 1

a. Use three different methods to solve tan2 u 2 1 5 0 for tan u.

b. Find all possible values of u in the interval 0 � u , 2p.

Solution a. METHOD 1: FACTOR METHOD 2: SQUARE ROOT

tan2 u 2 1 5 0 tan 2 u 2 1 5 0

(tan u 1 1)(tan u 2 1) 5 0 tan2 u 5 1

tan u 1 1 5 0 tan u 2 1 5 0 tan u 5 61

tan u 5 21 tan u 5 1

METHOD 3: QUADRATIC FORMULA

tan2 u 2 1 5 0

a 5 1, b 5 0, c 5 21

tan u 5 5

b. When tan u 5 1, u is in quadrant I or in quadrant III.

In quadrant I: if tan u 5 1, u 5

This is also the measure of the reference angle.

In quadrant III: u 5 p 1

Answer

EXAMPLE 2

Find, to the nearest degree, all possible values of B such that:

3 sin2 B 1 3 sin B 2 2 5 0

Solution a 5 3, b 5 3, c 5 22

sin B 5 5

Let sin B 5 .

Since is a number between 21 and 1, it is in the range of the

sine function.

23 1 !336 < 0.46

23 1 !336CASE 1

23 6 !9 1 246 5

23 6 !336

23 6 "(3)2 2 4(3)(22)2(3)

Up4 , 3p

4 , 5p4 , 7p

4 V

p4 5 5p

4

p4

6!42 5 62

2 5 6120 6 "02 2 4(1)(21)

2(1)


When tan u 5 21, u is in quadrantII or in quadrant IV.

One value of u is . The refer-ence angle is .

In quadrant II: u 5

In quadrant IV: u 5 2p 2 p4 5 7p

4

p 2 p4 5 3p

4

p4

2p4

14411C13.pgs 8/12/08 1:56 PM Page 532

ENTER: 23 DISPLAY:

33

6

The sine function is positive in the first and second quadrants.

In quadrant I: B 5 27°In quadrant II: B 5 180 2 27 5 153°

Let sin B 5 .

Since 21.46 is not a number between 21 and 1, it is not in

the range of the sine function. There are no values of B such that

sin B 5 . ✘

Enter Y1 5 3 sin2 X 1 3 sin X 2 2 into the menu.

ENTER: 3 3 2

With the calculator set to DEGREE mode, graphthe function in the following viewing window:

Xmin 5 0, Xmax 5 360, Xscl 5 30, Ymin 5 25,Ymax 5 5

The solutions are the x-coordinates of the x-intercepts, that is, the roots of Y1. Use the zerofunction of your graphing calculator to find the roots. Press . Use the arrows to enter a left bound to the left of one of the zero values, a right bound to theright of the zero value, and a guess near the zero value. The calculator will dis-play the coordinates of the point at which the graph intersects the x-axis.Repeat to find the other root.

As before, we find that the solutions in the interval 0° � u , 360° are approxi-mately 27° and 153°.

Answer B 5 27° 1 360n or B 5 153° 1 360n for integral values of n.

Z e r oX = 1 5 2 . 7 7 8 7 9 Y = 0

*Z e r oX = 2 7 . 2 2 1 2 0 8 Y = 0

*

2CALC2nd

�)X,T,�,nSIN�x2)X,T,�,nSINY�

Y�CalculatorSolution

23 2 !336

23 2 !336 <

23 2 !336CASE 2

ENTER)�

))�¯2nd 2 7 . 2 2 1 2 0 7 6 8

s i n - 1 ( ( - 3 + √ ( 3 3 ) )/ 6 )

�(SIN�12nd

Using the Quadratic Formula to Solve Trigonometric Equations 533

14411C13.pgs 8/12/08 1:56 PM Page 533


1. The discriminant of the quadratic equation tan2 u 1 4 tan u 1 5 5 0 is 24. Explain why thesolution set of this equation is the empty set.

2. Explain why the solution set of 2 csc2 u 2 csc u 5 0 is the empty set.

Developing SkillsIn 3–14, use the quadratic formula to find, to the nearest degree, all values of u in the interval 0° � u , 360° that satisfy each equation.

3. 3 sin2 u 2 7 sin u 2 3 5 0 4. tan2 u 2 2 tan u 2 5 5 0

5. 7 cos2 u 2 1 5 5 cos u 6. 9 sin2 u 1 6 sin u 5 2

7. tan2 u 1 3 tan u 1 1 5 0 8. 8 cos2 u 2 7 cos u 1 1 5 0

9. 2 cot2 u 1 3 cot u 2 4 5 0 10. sec2 u 2 2 sec u 2 4 5 0

11. 3 csc2 u 2 2 csc u 5 2 12. 2 tan u (tan u 1 1) 5 3

13. 3 cos u 1 1 5 14.

15. Find all radian values of u in the interval 0 � u , 2p for which .

16. Find, to the nearest hundredth of a radian, all values of u in the interval 0 � u , 2p forwhich .

When an equation contains two different functions, it may be possible to factorin order to write two equations, each with a different function. We can also useidentities to write an equivalent equation with one function.

The equation cos2 u 1 sin u 5 1 cannot be solved by factoring. We can usethe identity cos2 u 1 sin2 u 5 1 to change the equation to an equivalent equa-tion in sin u by replacing cos2 u with 1 2 sin2 u.

cos2 u 1 sin u 5 1

1 2 sin2 u 1 sin u 5 1

2sin2 u 1 sin u 5 0

sin u (2sin u 1 1) 5 0

sin u 5 0 2sin u 1 1 5 0

u15 0° 1 5 sin u

u2 5 180° u3 5 90°

13-4 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC EQUATIONS INVOLVING MORE THAN ONE FUNCTION

cos u3 5 1

3 cos u 1 1

sin u1 5 1

2 sin u

sin u2 5 3

sin u 1 21

cos u

Exercises


14411C13.pgs 8/12/08 1:56 PM Page 534

Check u1 5 0° Check u2 5 180° Check u3 5 90°

✔ ✔ ✔

In the interval 0° � u , 360°, the solution set of cos2 u 1 sin u 5 1 is {0°, 90°, 180°}.

Is it possible to solve the equation cos2 u 1 sin u 5 1 by writing an equiva-lent equation in terms of cos u? To do so we must use an identity to write sin uin terms of cos u. Since sin2 u 5 1 2 cos2 u, sin u 5 .

(1) Write the equation: cos2 u 1 sin u 5 1

(2) Replace sin u with cos2 u 5 1

:

(3) Isolate the radical: 5 1 2 cos2 u

(4) Square both sides of 1 2 cos2 u 5 1 2 2 cos2 u 1 cos4 u

the equation:

(5) Write an equivalent equation with the right cos2 u 2 cos4 u 5 0side equal to 0:

(6) Factor the left side: cos2 u (1 2 cos2 u) 5 0

(7) Set each factor equal to 0: cos2 u 5 0 1 2 cos2 u 5 0

cos u 5 0 1 5 cos2 u

u15 90° 61 5 cos u

u2 5 270° u3 5 0°

u4 5 180°

This approach uses more steps than the first. In addition, because it involvessquaring both sides of the equation, an extraneous root, 270°, has been intro-duced. Note that 270° is a root of the equation cos2 u 2 cos4 u 5 0 but is not aroot of the given equation.

cos2 u 1 sin u 5 1

cos2 270° 1 sin 270° 1

(0)2 1 (21) 1

0 2 1 � 1 ✘

5?

5?

6"1 2 cos2 u

6"1 2 cos2 u

6"1 2 cos2 u

6"1 2 cos2 u

1 5 11 5 11 5 1

02 1 1 5? 1(21)2 1 0 5? 112 1 0 5? 1

cos 2

908 1 sin 908 5? 1 cos 2

1808 1 sin 1808 5? 1 cos 2

08 1 sin 08 5? 1

cos 2

u 1 sin u 5 1 cos 2

u 1 sin u 5 1 cos 2

u 1 sin u 5 1

Using Substitution to Solve Trigonometric Equations Involving More than One Function 535

14411C13.pgs 8/12/08 1:56 PM Page 535

Any of the eight basic identities or the related identities can be substitutedin a given equation.

EXAMPLE 1

Find all values of A in the interval 0° � A , 360° such that 2 sin A 1 1 5 csc A.


(1) Write the equation: 2 sin A 1 1 5 csc A

(2) Replace csc A with : 2 sin A 1 1 5

(3) Multiply both sides of the 2 sin2 A 1 sin A 5 1equation by sin A:

(4) Write an equivalent equation 2 sin2 A 1 sin A 2 1 5 0with 0 as the right side:

(5) Factor the left side: (2 sin A 2 1)(sin A 1 1) 5 0

(6) Set each factor equal to 0 2 sin A 2 1 5 0 sin A 1 1 5 0and solve for sin A: 2 sin A 5 1 sin A 5 21

sin A 5 A 5 270°

A 5 30°

or A 5 150°

Answer A 5 30° or A 5 150° or A 5 270°

Often, more than one substitution is necessary to solve an equation.

EXAMPLE 2

If 0 � u , 2p, find the solution set of the equation 2 sin u 5 3 cot u.

Solution How to Proceed(1) Write the equation: 2 sin u 5 3 cot u

(2) Replace cot u with : 2 sin u 5 3

(3) Multiply both sides of the 2 sin2 u 5 3 cos uequation by sin u:

(4) Replace sin2 u with 1 2 cos2 u: 2(1 2 cos2 u) 5 3 cos u

(5) Write an equivalent equation 2 2 2 cos2 u 5 3 cos uin standard form:

A cos usin u Bcos u

sin u

12

1sin A

1sin A


2 2 2 cos2 u 2 3 cos u 5 0

2 cos2 u 1 3 cos u 2 2 5 0

14411C13.pgs 8/12/08 1:56 PM Page 536

(6) Factor and solve for cos u: (2 cos u 2 1)(cos u 1 2) 5 0

(7) Find all values of u in the 2 cos u 2 1 5 0 cos u 1 2 5 0given interval: 2 cos u 5 1 cos u 5 22 ✘

cos u 5 No solution

u 5

or u 5

Answer

The following identities from Chapter 10 will be useful in solving trigono-metric equations:

Up3 , 5p

3 V

5p3

p3

12

Using Substitution to Solve Trigonometric Equations Involving More than One Function 537

Reciprocal Identities Quotient Identities Pythagorean Identities

sec u 5 tan u 5 cos2 u 1 sin2 u 5 1

csc u 5 cot u 5 1 1 tan2 u 5 sec2 u

cot u 5 cot2 u 1 1 5 csc2 u1tan u

cos usin u

1sin u

sin ucos u

1cos u


1. Sasha said that sin u 1 cos u 5 2 has no solution. Do you agree with Sasha? Explain why orwhy not.

2. For what values of u is sin u 5 true?

Developing SkillsIn 3–14, find the exact values of u in the interval 0° � u , 360° that satisfy each equation.

3. 2 cos2 u 2 3 sin u 5 0 4. 4 cos2 u 1 4 sin u 2 5 5 0

5. csc2 u 2 cot u 2 1 5 0 6. 2 sin u 1 1 5 csc u

7. 2 sin2 u 1 3 cos u 2 3 5 0 8. 3 tan u 5 cot u

9. 2 cos u 5 sec u 10. sin u 5 csc u

11. tan u 5 cot u 12. 2 cos2 u 5 sin u 1 2

13. cot2 u 5 csc u 1 1 14. 2 sin2 u 2 tan u cot u 5 0

"1 2 cos2 u

Exercises

14411C13.pgs 8/12/08 1:56 PM Page 537

Applying Skills15. An engineer would like to model a piece for a factory

machine on his computer. As shown in the figure, themachine consists of a link fixed to a circle at point A.The other end of the link is fixed to a slider at pointB. As the circle rotates, point B slides back and forthbetween the two ends of the slider (C and D). Themovement is restricted so that u, the measure of�AOD, is in the interval 245° � u � 45°. The motionof point B can be described mathematically by theformula

where r is the radius of the circle and l is the length of the link. Both the radius of the circleand the length of the link are 2 inches.

a. Find the exact value of CB when: (1) u 5 30° (2) u 5 45°.

b. Find the exact value(s) of u when CB 5 2 inches.

c. Find, to the nearest hundredth of a degree, the value(s) of u when CB 5 1.5 inches.

If an equation contains function values of two different but related angle mea-sures, we can use identities to write an equivalent equation in terms of just onevariable. For example: Find the value(s) of u such that sin 2u 2 sin u 5 0.

Recall that sin 2u 5 2 sin u cos u. We can use this identity to write the equa-tion in terms of just one variable, u, and then use any convenient method tosolve the equation.

(1) Write the equation: sin 2u 2 sin u 5 0

(2) For sin 2u, substitute its equal, 2 sin u cos u 2 sin u 5 02 sin u cos u:

(3) Factor the left side: sin u (2 cos u 2 1) 5 0

(4) Set each factor equal to zero sin u 5 0 2 cos u 2 1 5 u

and solve for u: u 5 0° 2 cos u 5 1

or u 5 180° cos u 5

u 5 60°

or u 5 300°

12

13-5 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC EQUATIONS INVOLVING DIFFERENT ANGLE MEASURES

CB 5 r(cos u 2 1) 1 "l2 2 r2 sin2

u


O

A

C

u

B D

14411C13.pgs 8/12/08 1:56 PM Page 538

The degree measures 0°, 60°, 180°, and 300° are all of the values in the inter-val 0° � u , 360° that make the equation true.Any values that differ from thesevalues by a multiple of 360° will also make the equation true.

EXAMPLE 1

Find, to the nearest degree, the roots of cos 2u 2 2 cos u 5 0.


(1) Write the given equation:

(2) Use an identity to writecos 2u in terms of cos u:

(3) Write the equation in standardform:

(4) The equation cannot be factored over the set of integers. Use the quadratic formula:

(5) When we use a calculator to approximate the value of

arccos , the calculator will return the value 111°. Cosine is negative in both the second and the third quadrants. Therefore,there is both a second-quadrant and a third-quadrant angle such

that cos u 5 :

(6) When we use a calculator to approximate the value of

arccos , the calculator will return an error message because

. 1 is not in the domain

of arccosine:

Answer To the nearest degree, u 5 111° or u 5 249°.

1 1 !32

1 1 !32

1 2 !32

1 2 !32

Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures 539

cos 2u 2 2 cos u 5 0

2 cos2 u 2 1 2 2 cos u 5 0

2 cos2 u 2 2 cos u 2 1 5 0

cos u 5

5

51 6 !3

2

2 6 !124

2(22) 6 "(22)2 2 4(2)(21)2(2)

Q u i t2 : G o t o

E R R : D O M A I N1 :

u1 5 111°

R 5 180° 2 111° 5 69°

u2 5 180° 1 69° 5 249°

1 1 1 . 4 7 0 7 0 1 4c o s - 1 ( ( 1 - √ ( 3 ) ) / 2 )

14411C13.pgs 8/12/08 1:56 PM Page 539


EXAMPLE 2

Find, to the nearest degree, the values of u in the interval 0° � u � 360° that aresolutions of the equation sin (90° 2 u) 1 2 cos u 5 2.

Solution Use the identity sin (90° 2 u) 5 cos u.

sin(90° 2 u) 1 2 cos u 5 2

cos u 1 2 cos u 5 2

3 cos u 5 2

cos u 5

To the nearest degree, a calculator returns the value of u as 48°.

In quadrant I, u 5 48° and in quadrant IV, u 5 360° 2 48° 5 312°.

Answer u 5 48° or u 5 312°

The basic trigonometric identities along with the cofunction, double-angle,and half-angle identities will be useful in solving trigonometric equations:

Note that in radians, the right sides of the cofunction identities are writtenin terms of p 2 u.

Writing About Mathematics1. Isaiah said that if the equation cos 2x 1 2 cos2 x 5 2 is divided by 2, an equivalent equation

is cos x 1 cos2 x 5 1. Do you agree with Isaiah? Explain why or why not.

2. Aaron solved the equation 2 sin u cos u 5 cos u by first dividing both sides of the equation

by cos u. Aaron said that for 0 � u � 2p, the solution set is . Do you agree with Aaron? Explain why or why not.

Up6 , 5p

6 V

Exercises

23

Cofunction Identities Double-Angle Identities Half-Angle Identities

cos u 5 sin (90° 2 u) sin (2u) 5 2 sin u cos u sin 5

sin u 5 cos (90° 2 u) cos (2u) 5 cos2 u 2 sin2 u cos 5

tan u 5 cot (90° 2 u) tan (2u) 5 tan 5

cot u 5 tan (90° 2 u)

sec u 5 csc (90° 2 u)

csc u 5 sec (90° 2 u)

6#1 2 cos u1 1 cos u

12u2 tan u

1 2 tan2 u

6#1 1 cos u2

12u

6#1 2 cos u2

12u

14411C13.pgs 8/12/08 1:56 PM Page 540

Developing SkillsIn 3–10, find the exact values of u in the interval 0° � u � 360° that make each equation true.

3. sin 2u 2 cos u 5 0 4. cos 2u 1 sin2 u 5 1

5. sin 2u 1 2 sin u 5 0 6. tan 2u 5 cot u

7. cos 2u 1 2 cos2 u 5 2 8. sin 5 cos u

9. 3 2 3 sin u 2 2 cos2 u 5 0 10. 3 cos 2u 2 4 cos2 u 1 2 5 0

In 11–18, find all radian measures of u in the interval 0 � u � 2p that make each equation true.Express your answers in terms of p when possible; otherwise, to the nearest hundredth.

11. cos 2u 5 2 cos u 2 2 cos2 u 12. 2 sin 2u 1 sin u 5 0

13. 5 sin2 u 2 4 sin u 1 cos 2u 5 0 14. cos u 5 3 sin 2u

15. 3 sin 2u 5 tan u 16. sin (90° 2 u) 1 cos2 u 5

17. 2 cos2 u 1 3 sin u 2 2 cos 2u 5 1 18. (2 sin u cos u)2 1 4 sin 2u 2 1 5 0

Applying Skills

19. Martha swims 90 meters from point A on thenorth bank of a stream to point B on the oppositebank. Then she makes a right angle turn andswims 60 meters from point B to point C, anotherpoint on the north bank. If m�CAB 5 u, thenm�ACB 5 90° 2 u.

a. Let d be the width of the stream, the length ofthe perpendicular distance from B to .Express d in terms of sin u.

b. Express d in terms of sin (90° 2 u).

c. Use the answers to a and b to write an equation. Solve the equation for u.

d. Find d, the width of the stream.

20. A pole is braced by two wires of equal length asshown in the diagram. One wire, , makes an angleof u with the ground, and the other wire, , makesan angle of 2u with the ground. If FD 5 1.75FB, find,to the nearest degree, the measure of u:

a. Let AB = CD = x, FB 5 y, and FD 5 1.75y.Express sin u and sin 2u in terms of x and y.

b. Write an equation that expresses a relationshipbetween sin u and sin 2u and solve for u to thenearest degree.

CDAB

AC

14

12u

Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures 541

u90° 2 u

C A

B

d

F ACu

B

D

2u

14411C13.pgs 8/12/08 1:56 PM Page 541

CHAPTER SUMMARY

A trigonometric equation is an equation whose variable is expressed interms of a trigonometric function value.

The following table compares the degree measures of u from 290° to 360°, the radian measures of u from 2 to 2p, and the measure of its referenceangle.

To solve a trigonometric equation:

1. If the equation involves more than one variable, use identities to write theequation in terms of one variable.

2. If the equation involves more than one trigonometric function of the samevariable, separate the functions by factoring or use identities to write theequation in terms of one function of one variable.

3. Solve the equation for the function value of the variable. Use factoring orthe quadratic formula to solve a second-degree equation.

4. Use a calculator or your knowledge of the exact function values to writeone value of the variable to an acceptable degree of accuracy.

5. If the measure of the angle found in step 4 is not that of a quadrantalangle, find the measure of its reference angle.

6. Use the measure of the reference angle to find the degree measures ofeach solution in the interval 0° � u , 360° or the radian measures of eachsolution in the interval 0 � u , 2p.

7. Add 360n (n an integer) to the solutions in degrees found in steps 4 and 6to write all possible solutions in degrees. Add 2pn (n an integer) to thesolutions in radians found in steps 2 and 4 to write all possible solutions inradians.

VOCABULARY

13-1 Trigonometric equation

p2


Fourth First Second Third FourthQuadrant Quadrant Quadrant Quadrant Quadrant

Angle 290° , u , 0° 0° , u , 90° 90° , u , 180° 180° , u , 270° 270° , u , 360°, u , 0 , u , u p , u , , u , 2p

Reference 2u u 180° 2 u u 2 180° 360° 2 u

Angle 2u u p 2 u u 2 p 2p 2 u

3p2

3p2

p20 , u , p

22p2

14411C13.pgs 8/12/08 1:56 PM Page 542

REVIEW EXERCISES

In 1–10, find the exact values of x in the interval 0° � x � 360° that make eachequation true.

1. 2 cos x 1 1 5 0 2. 2 sin x 5 sin x 1

3. 2 sec x 5 2 1 sec x 4. 2 cos2 x 1 cos x 2 1 5 0

5. cos x sin x 1 sin x 5 0 6. tan x 2 3 cot x 5 0

7. 2 cos x 2 sec x 5 0 8. sin2 x 2 cos2 x 5 0

9. 2 tan x 5 10. cos3 x 2 5 0

In 11–22, find, to the nearest hundredth, all values of u in the interval 0 � u � 2p that make each equation true.

11. 7 sin u 1 3 5 1 12. 5(cos u 2 1) 5 6 1 cos u

13. 4 sin2 u 2 3 sin u 5 1 14. 3 cos2 u 2 cos u 2 2 5 0

15. tan2 u 2 4 tan u 2 1 5 0 16. sec2 u 2 10 sec u 1 20 5 0

17. tan 2u 5 4 tan u 18. 2 sin 2u 1 cos u 5 0

19. 5 4 20. 3 cos 2u 1 cos u 1 2 5 0

21. 2 tan2 u 1 6 tan u 5 20 22. cos 2u 2 cos2 u 1 cos u 1 5 0

23. Explain why the solution set of tan u 2 sec u 5 0 is the empty set.

24. In �ABC, m�A 5 u and m�B 5 2u. Thealtitude from C intersects at D andAD : DB 5 5 : 2.

a. Write tan u and tan 2u as ratios of thesides of �ADC and �BDC, respec-tively.

b. Solve the equation for tan 2u found ina for CD.

c. Substitute the value of CD found in b into the equation of tan u foundin a.

d. Solve for u.

e. Find the measures of the angles of the triangle to the nearest degree.

AB

14

sin 2u1 1 cos 2u

34 cos x1 2 tan2 x

!12!3

Review Exercises 543

A

C

u BD

2u

14411C13.pgs 8/12/08 1:56 PM Page 543

SECTION 8.8 vectors 729

leARnInG OBjeCTIVeS

In this section, you will:• View vectors geometrically.• Find magnitude and direction.• Perform vector addition and scalar multiplication.• Find the component form of a vector.• Find the unit vector in the direction of v.• Perform operations with vectors in terms of i and j.• Find the dot product of two vectors.

8.8 VeCTORS

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in Figure 1. What are the ground speed and actual bearing of the plane?

Figure 1

N

A

O

BX C

200

140˚

16.2

α

Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

A Geometric View of VectorsA vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:

• Lower case, boldfaced type, with or without an arrow on top such as v, u, w, v, u, w.

• Given initial point P and terminal point Q, a vector can be represented as PQ. The arrowhead on top is what indicates that it is not just a line, but a directed line segment.

• Given an initial point of (0, 0) and terminal point (a, b), a vector may be represented as 〈a, b⟩.

This last symbol 〈a, b⟩ has special significance. It is called the standard position. The position vector has an initial point (0, 0) and a terminal point 〈a, b⟩. To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector CD is C(x1, y1) and the terminal point is D(x2, y2), then the position vector is found by calculating

AB = 〈x2 − x1, y2 − y1⟩

= 〈a, b⟩

→→→

→

→

→

CHAPTER 8 Further applications oF trigonometry730

In Figure 2, we see the original vector CD and the position vector AB.

Figure 2

2

–1–1–2

–2

–3

–3

–4

–4–5

1

3

3

21 4

4

5 6 7 8

5

x

D

A

C

y

(x2, y2)

(x1, y1)B

(a, b)

properties of vectors A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at (0, 0) and is identified by its terminal point 〈a, b⟩.

Example 1 Find the Position Vector

Consider the vector whose initial point is P(2, 3) and terminal point is Q(6, 4). Find the position vector.

Solution The position vector is found by subtracting one x-coordinate from the other x-coordinate, and one y-coordinate from the other y-coordinate. Thus v = 〈6 −2, 4 −3⟩

= 〈4, 1⟩

The position vector begins at (0, 0) and terminates at (4, 1). The graphs of both vectors are shown in Figure 3.

Figure 3

2

–1–1–2

–2

–3

–3

–4

–4–5

1

3

3

210 4

4

5 6 7 8

5

x

y

P (2, 3)

Q (6, 4)

⟨4, 1⟩

We see that the position vector is 〈4, 1⟩.

Example 2 Drawing a Vector with the Given Criteria and Its Equivalent Position Vector

Find the position vector given that vector v has an initial point at (−3, 2) and a terminal point at (4, 5), then graph both vectors in the same plane.

Solution The position vector is found using the following calculation:

v = 〈4 − ( − 3), 5 − 2⟩

= 〈7, 3⟩

Thus, the position vector begins at (0, 0) and terminates at (7, 3). See Figure 4.

→ →


Figure 4

2

–1–1–2

–2

–3

–3

–4

–4–5

1

3

3

21 4

4

5 6 7 8

5

x

y

(7, 3)

(4, 5)

(−3, 2)

(0, 0) Position vector

Try It #1

Draw a vector v that connects from the origin to the point (3, 5).

Finding Magnitude and DirectionTo work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.

magnitude and direction of a vector

Given a position vector v = 〈a, b⟩, the magnitude is found by | v | = √—

a 2 + b 2 . The direction is equal to the angle formed with the x-axis, or with the y-axis, depending on the application. For a position vector, the direction is

found by tan θ = b __ a ⇒ θ = tan −1 b __ a , as illustrated in Figure 5.

Figure 5

2

– 1– 1

1

3

3

21 4

4

5

5

x

y

|v|

θ

⟨a, b⟩

Two vectors v and u are considered equal if they have the same magnitude and the same direction. Additionally, if both vectors have the same position vector, they are equal.

Example 3 Finding the Magnitude and Direction of a Vector

Find the magnitude and direction of the vector with initial point P(−8, 1) and terminal point Q(−2, −5). Draw the vector.

Solution First, find the position vector.

u = 〈−2, − (−8), −5−1⟩

= 〈6, − 6⟩

We use the Pythagorean Theorem to find the magnitude.

| u | = √—

(6) 2 + (−6) 2

= √—

72

= 6 √— 2


The direction is given as

tan θ = −6 ___ 6 = −1 ⇒ θ = tan −1 (−1)

= −45°

However, the angle terminates in the fourth quadrant, so we add 360° to obtain a positive angle. Thus, −45° + 360° = 315°. See Figure 6.

Figure 6

2

–1–1–2

–2

–3

–3

–4

–4

–5–6

–5–6

1

3

3

21 4

4

5 6

56

315°x

y

Example 4 Showing That Two Vectors Are Equal

Show that vector v with initial point at (5, −3) and terminal point at (−1, 2) is equal to vector u with initial point at (−1, −3) and terminal point at (−7, 2). Draw the position vector on the same grid as v and u. Next, find the magnitude and direction of each vector.

Solution As shown in Figure 7, draw the vector v starting at initial (5, −3) and terminal point (−1, 2). Draw the vector u with initial point (−1, −3) and terminal point (−7, 2). Find the standard position for each.

Next, find and sketch the position vector for v and u. We have

v = 〈−1 − 5, 2 − ( − 3)⟩

= 〈−6, 5⟩

u = 〈−7 − (−1), 2 − (−3)⟩

= 〈−6, 5⟩

Since the position vectors are the same, v and u are the same.

An alternative way to check for vector equality is to show that the magnitude and direction are the same for both vectors. To show that the magnitudes are equal, use the Pythagorean Theorem.

| v | = √——

(−1 − 5) 2 + (2 − (−3)) 2

= √—

(−6)2 + (5)2

= √—

36 + 25

= √—

61

| u | = √——

(−7 − (−1))2 + (2 − (−3))2

= √—

(−6)2 + (5)2

= √—

36 + 25

= √—

61

As the magnitudes are equal, we now need to verify the direction. Using the tangent function with the position vector gives tan θ = − 5 _

6 ⇒ θ = tan−1 − 5 _

6

= −39.8°

However, we can see that the position vector terminates in the second quadrant, so we add 180°. Thus, the direction is −39.8° + 180° = 140.2°.


Figure 7

2

x

vu

Position vector

y

–1–1–2

–2

–3

–3

–4

–4

–5–6–7

–5

1

3

3

21 4

4

5

5

140.2°

Performing Vector Addition and Scalar MultiplicationNow that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector u = 〈x, y⟩ as an arrow or directed line segment from the origin to the point (x, y), vectors can be situated anywhere in the plane. The sum of two vectors u and v, or vector addition, produces a third vector u+ v, the resultant vector.

To find u + v, we first draw the vector u, and from the terminal end of u, we drawn the vector v. In other words, we have the initial point of v meet the terminal end of u. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum u + v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown in Figure 8.

Figure 8

−vvu

uu − v

u + v

Vector subtraction is similar to vector addition. To find u − v, view it as u + (−v). Adding −v is reversing direction of v and adding it to the end of u. The new vector begins at the start of u and stops at the end point of −v. See Figure 9 for a visual that compares vector addition and vector subtraction using parallelograms.

Figure 9

u

v

u + v

–v

u

u

u – v

Example 5 Adding and Subtracting Vectors

Given u = 〈3, − 2⟩ and v = 〈−1, 4⟩, find two new vectors u + v, and u − v.

Solution To find the sum of two vectors, we add the components. Thus, u + v = 〈3, − 2⟩ + 〈−1, 4⟩ = 〈3 + ( − 1), − 2 + 4⟩ = 〈2, 2⟩

See Figure 10(a).

To find the difference of two vectors, add the negative components of v to u. Thus, u + (−v) = 〈3, − 2⟩ + 〈1, −4⟩ = 〈3 + 1, − 2 + (−4)⟩ = 〈4, − 6⟩


See Figure 10(b).

Figure 10 (a) Sum of two vectors (b) Difference of two vectors

2

xu

v

u + v

y

–1–1–2

–2

–3

–3

–4

–4

–5

1

3

3

21 4

4

5

(a)

− v

(b)

2

xu

u − v

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5–6

1

321 4 5

Multiplying By a ScalarWhile adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

scalar multiplication Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply v = 〈a, b⟩ by k, we have

kv = 〈ka, kb⟩

Only the magnitude changes, unless k is negative, and then the vector reverses direction.

Example 6 Performing Scalar Multiplication

Given vector v = 〈3, 1⟩ , find 3v, 1 _ 2

v, and −v.

Solution See Figure 11 for a geometric interpretation. If v = 〈3, 1⟩, then

3v = 〈3 ⋅ 3, 3 ⋅ 1⟩

= 〈9, 3⟩

1 _ 2

v = 〈  1 _ 2

⋅ 3, 1 _ 2

⋅ 1⟩ = 〈  3 _

2 , 1 _

2 ⟩

−v = 〈−3, −1⟩

Figure 11

3v

v

–vv1

2

Analysis Notice that the vector 3v is three times the length of v, 1 _ 2

v is half the length of v, and −v is the same length of v, but in the opposite direction.


Try It #2

Find the scalar multiple 3u given u = 〈5, 4⟩ .

Example 7 Using Vector Addition and Scalar Multiplication to Find a New Vector

Given u = 〈3, − 2⟩ and v = 〈−1, 4⟩, find a new vector w = 3u + 2v.

Solution First, we must multiply each vector by the scalar.

3u = 3 〈3, − 2⟩

= 〈9, − 6⟩

2v = 2 〈−1, 4⟩

= 〈−2, 8⟩

Then, add the two together.

w = 3u + 2v

= 〈9, − 6⟩ + 〈−2, 8⟩

= 〈9 − 2, − 6 + 8⟩

= 〈7, 2⟩

So, w = 〈7, 2⟩.

Finding Component FormIn some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the x direction, and the vertical component is the y direction. For example, we can see in the graph in Figure 12 that the position vector 〈2, 3⟩ comes from adding the vectors v1 and v2. We have v1 with initial point (0, 0) and terminal point (2, 0).

v1 = 〈2 − 0, 0 − 0⟩

= 〈2, 0⟩

We also have v2 with initial point (0, 0) and terminal point (0, 3).

v2 = 〈0 − 0, 3 − 0⟩

= 〈0, 3⟩

Therefore, the position vector is v = 〈2 + 0, 3 + 0⟩

= 〈2, 3⟩

Using the Pythagorean Theorem, the magnitude of v1 is 2, and the magnitude of v2 is 3. To find the magnitude of v, use the formula with the position vector. | v | = √

__________

∣ v1 ∣ 2 + ∣ v2 ∣ 2

= √—

22 + 32

= √—

13

The magnitude of v is √—

13 . To find the direction, we use the tangent function tan θ = y _ x .

tan θ = v2 _ v1

tan θ = 3 _ 2

θ = tan−1 3 _ 2

= 56.3°


Figure 12

1

x

y

–1

–1

1 2

2

3 4

3

4

56.3° (2, 0)v1

v2|v|

Thus, the magnitude of v is √—

13 and the direction is 56.3° off the horizontal.

Example 8 Finding the Components of the Vector

Find the components of the vector v with initial point (3, 2) and terminal point (7, 4).

Solution First find the standard position.

v = 〈7 − 3, 4 − 2⟩

= 〈4, 2⟩

See the illustration in Figure 13.

Figure 13

1

x

y

–1

–1

1 2

2

3 4

3

4

v2 = ⟨0, 2⟩

v1 = ⟨4, 0⟩

The horizontal component is v1 = 〈4, 0⟩ and the vertical component is v2 = 〈0, 2⟩.

Finding the Unit Vector in the Direction of vIn addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.

Unit vectors are defined in terms of components. The horizontal unit vector is written as i = 〈1, 0⟩ and is directed along the positive horizontal axis. The vertical unit vector is written as j = 〈0, 1⟩ and is directed along the positive vertical axis. See Figure 14.

Figure 14

1

x

y

–1

–1

1 2

2

3 4

3

4

j = ⟨0, 1⟩i = ⟨1, 0⟩


the unit vectors If v is a nonzero vector, then v _

∣ v ∣ is a unit vector in the direction of v. Any vector divided by its magnitude is a

unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.

Example 9 Finding the Unit Vector in the Direction of v

Find a unit vector in the same direction as v = 〈−5, 12⟩.

Solution First, we will find the magnitude. | v | = √

— (−5)2 + (12)2

= √—

25 + 144

= √—

169

= 13

Then we divide each component by | v | , which gives a unit vector in the same direction as v:

v _ | v | = − 5 _ 13

i + 12 _ 13

j

or, in component form v _ | v | = 〈− 5 _

13 , 12 _

13 ⟩

See Figure 15.

Figure 15

2

x

y

–1–1–2–3–4–5–6–7–8

1

3

210

456789

1011121314

5 121313

,−

⟨−5, 12⟩

Verify that the magnitude of the unit vector equals 1. The magnitude of − 5 _ 13

i + 12 _ 13

j is given as

√ __________________

− 5 ___ 13 2 + 12 ___ 13

2 = √

_________

25 ____ 169  + 144 ___ 169

= √____

169 _ 169

= 1

The vector u = 5 ___ 13

i + 12 ___ 13

j is the unit vector in the same direction as v = 〈−5, 12⟩.

Performing Operations with Vectors in Terms of i and jSo far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of i and j.


vectors in the rectangular plane Given a vector v with initial point P = (x1, y1) and terminal point Q = (x2, y2), v is written as

v = (x2 − x1)i + (y1 − y2) j

The position vector from (0, 0) to (a, b), where (x2 − x1) = a and (y2 − y1) = b, is written as v = ai + bj. This vector sum is called a linear combination of the vectors i and j.

The magnitude of v = ai + bj is given as | v | = √—

a2 + b2 . See Figure 16.

Figure 16

v = ai + bjbj

ai

Example 10 Writing a Vector in Terms of i and j

Given a vector v with initial point P = (2, −6) and terminal point Q = (−6, 6), write the vector in terms of i and j.

Solution Begin by writing the general form of the vector. Then replace the coordinates with the given values.

v = (x2 − x1)i + (y2 − y1) j

= ( −6 − 2)i + (6 − ( − 6)) j

= − 8i + 12 j

Example 11 Writing a Vector in Terms of i and j Using Initial and Terminal Points

Given initial point P1 = (−1, 3) and terminal point P2 = (2, 7), write the vector v in terms of i and j.

Solution Begin by writing the general form of the vector. Then replace the coordinates with the given values.

v = (x2 − x1)i + (y2 − y1) j

v = (2 − ( − 1))i + (7 − 3) j

= 3i + 4 j

Try It #3

Write the vector u with initial point P = (−1, 6) and terminal point Q = (7, − 5) in terms of i and j.

Performing Operations on Vectors in Terms of i and jWhen vectors are written in terms of i and j, we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

adding and subtracting vectors in rectangular coordinates Given v = ai + bj and u = ci + dj, then v + u = (a + c)i + (b + d)j v − u = (a − c)i + (b − d)j


Example 12 Finding the Sum of the Vectors

Find the sum of v1 = 2i − 3j and v2 = 4i + 5j.

Solution According to the formula, we have

v1 + v2 = (2 + 4)i + ( − 3 + 5) j

= 6i + 2 j

Calculating the Component Form of a Vector: DirectionWe have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using i and j. For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.

Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with |v| replacing r.

vector components in terms of magnitude and direction Given a position vector v = 〈x, y⟩ and a direction angle θ,

cos θ = x _ | v | and sin θ = y _ | v |

x = | v | cos θ y = | v | sin θ

Thus, v = xi + yj = |v|cos θi + |v|sin θj, and magnitude is expressed as |v| = √—

x2 + y2 .

Example 13 Writing a Vector in Terms of Magnitude and Direction

Write a vector with length 7 at an angle of 135° to the positive x-axis in terms of magnitude and direction.

Solution Using the conversion formulas x = |v| cos θi and y =|v| sin θ j, we find that

x = 7cos(135°)i

= − 7 √— 2 _

2

y = 7sin(135°) j

= 7 √— 2 _

2

This vector can be written as v = 7cos(135°)i + 7sin(135°) j or simplified as

v = − 7 √— 2 _

2 i + 7 √

— 2 _

2 j

Try It #4

A vector travels from the origin to the point (3, 5). Write the vector in terms of magnitude and direction.

Finding the Dot Product of Two VectorsAs we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.

The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.


dot product The dot product of two vectors v = 〈a, b⟩ and u = 〈c, d⟩ is the sum of the product of the horizontal components and the product of the vertical components.

v ⋅ u = ac + bd

To find the angle between the two vectors, use the formula below.

cos θ = v ___ | v | ⋅ u ___ | u |

Example 14 Finding the Dot Product of Two Vectors

Find the dot product of v = 〈5, 12⟩ and u = 〈−3, 4⟩.

Solution Using the formula, we have v ⋅ u = 〈5, 12⟩ ⋅ 〈−3, 4⟩ = 5 ⋅ ( −3) + 12 ⋅ 4 = −15 + 48 = 33

Example 15 Finding the Dot Product of Two Vectors and the Angle between Them

Find the dot product of v1 = 5i + 2j and v2 = 3i + 7j. Then, find the angle between the two vectors.

Solution Finding the dot product, we multiply corresponding components. v1 ⋅ v2 = 〈5, 2⟩ ⋅ 〈3, 7⟩

= 5 ⋅ 3 + 2 ⋅ 7 = 15 + 14 = 29

To find the angle between them, we use the formula cos θ = v _ | v | ⋅ u _ | u |

v _ | v | ⋅ u _ | u | = 〈 5 ______

√—

29 + 2 ______

√—

29 ⟩ ⋅ 〈 3 ______

√—

58 + 7 ______

√—

58 ⟩

= 5 ______ √

— 29 ⋅ 3 ______

√—

58 + 2 ______

√—

29 ⋅ 7 ______

√—

58

= 15 ________ √—

1682 + 14 ________

√—

1682 = 29 ________

√—

1682

= 0.707107

cos−1(0.707107) = 45°

See Figure 17.

Figure 17

2

x

y

–1–1–2

–2–3

–3

–4

–4

–5–6

1

3

3

21 4

4

5 6

5678

45°


Example 16 Finding the Angle between Two Vectors

Find the angle between u = 〈−3, 4⟩ and v = 〈5, 12⟩.

Solution Using the formula, we have

θ = cos−1 u _ | u | ⋅ v _ | v |

u _ ∣ u ∣

⋅ v _ ∣ v ∣

= −3i + 4j

_ 5 ⋅ 5i + 12j

_ 13

= − 3 _ 5 ⋅ 5 _ 13

+ 4 _ 5 ⋅ 12 _ 13

= − 15 _ 65

+ 48 _ 65

= 33 _ 65

θ = cos−1 33 _ 65

= 59.5°

See Figure 18.

Figure 18

2

x

y

–1–1–2–3–4–5–6

1

3

3

21 4

4

5 6

56789

101112

59.5°

Example 17 Finding Ground Speed and Bearing Using Vectors

We now have the tools to solve the problem we introduced in the opening of the section.

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? See Figure 19.

Figure 19

N

A

O

BX C

200

140˚

16.2

α

Solution The ground speed is represented by x in the diagram, and we need to find the angle α in order to calculate the adjusted bearing, which will be 140° + α .


Notice in Figure 19, that angle BCO must be equal to angle AOC by the rule of alternating interior angles, so angle BCO is 140°. We can find x by the Law of Cosines:

x2 = (16.2)2 + (200)2 − 2(16.2)(200)cos(140°)

x2 = 45,226.41

x = √—

45, 226.41

x = 212.7

The ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.

sin α _ 16.2

= sin(140°)

_ 212.7

sin α = 16.2sin(140°)

__ 212.7

= 0.04896

sin−1(0.04896) = 2.8°

Therefore, the plane has a SE bearing of 140° + 2.8° = 142.8°. The ground speed is 212.7 miles per hour.

Access these online resources for additional instruction and practice with vectors.

• Introduction to Vectors (http://openstaxcollege.org/l/introvectors)

• Vector Operations (http://openstaxcollege.org/l/vectoroperation)

• The Unit Vector (http://openstaxcollege.org/l/unitvector)

SECTION 8.8 section exercises 743

8.8 SeCTIOn exeRCISeS

VeRBAl

1. What are the characteristics of the letters that are commonly used to represent vectors?

2. How is a vector more specific than a line segment?

3. What are i and j, and what do they represent? 4. What is component form?5. When a unit vector is expressed as 〈a, b⟩, which

letter is the coefficient of the i and which the j?

AlGeBRAIC

6. Given a vector with initial point (5, 2) and terminal point (−1, − 3), find an equivalent vector whose initial point is (0, 0). Write the vector in component form 〈a, b⟩.

7. Given a vector with initial point (−4, 2) and terminal point (3, − 3), find an equivalent vector whose initial point is (0, 0). Write the vector in component form 〈a, b⟩.

8. Given a vector with initial point (7, − 1) and terminal point (−1, − 7), find an equivalent vector whose initial point is (0, 0). Write the vector in component form 〈a, b⟩.

For the following exercises, determine whether the two vectors u and v are equal, where u has an initial point P1 and a terminal point P2 and v has an initial point P3 and a terminal point P4.

9. P1 = (5, 1), P2 = (3, − 2), P3 = (−1, 3), and P4 = (9, − 4)

10. P1 = (2, −3), P2 = (5, 1), P3 = (6, − 1), and P4 = (9, 3)

11. P1 = (−1, − 1), P2 = (−4, 5), P3 = (−10, 6), and P4 = (−13, 12)

12. P1 = (3, 7), P2 = (2, 1), P3 = (1, 2), and P4 = (−1, − 4)

13. P1 = (8, 3), P2 = (6, 5), P3 = (11, 8), and P4 = (9, 10) 14. Given initial point P1 = (−3, 1) and terminal point P2 = (5, 2), write the vector v in terms of i and j.

15. Given initial point P1 = (6, 0) and terminal point P2 = (−1, − 3), write the vector v in terms of i and j.

For the following exercises, use the vectors u = i + 5j, v = −2i − 3j, and w = 4i − j.

16. Find u + (v − w) 17. Find 4v + 2u

For the following exercises, use the given vectors to compute u + v, u − v, and 2u − 3v.

18. u = 〈2, − 3⟩ , v = 〈1, 5⟩ 19. u = 〈−3, 4⟩ , v = 〈−2, 1⟩20. Let v = −4i + 3j. Find a vector that is half the

length and points in the same direction as v.21. Let v = 5i + 2j. Find a vector that is twice the length

and points in the opposite direction as v.

For the following exercises, find a unit vector in the same direction as the given vector.

22. a = 3i + 4j 23. b = −2i + 5j 24. c = 10i − j 25. d = − 1 __ 3 i + 5 __ 2 j26. u = 100i + 200j 27. u = −14i + 2j

For the following exercises, find the magnitude and direction of the vector, 0 ≤ θ < 2π.

28. 〈0, 4⟩ 29. 〈6, 5⟩ 30. 〈2, −5⟩ 31. 〈−4, −6⟩

32. Given u = 3i − 4j and v = −2i + 3j, calculate u ⋅ v. 33. Given u = −i − j and v = i + 5j, calculate u ⋅ v.

34. Given u = 〈−2, 4⟩ and v = 〈−3, 1⟩, calculate u ⋅ v. 35. Given u = 〈−1, 6⟩ and v = 〈6, − 1⟩, calculate u ⋅ v.

744 CHAPTER 8 Further applications oF trigonometry

GRAPHICAl

For the following exercises, given v, draw v, 3v and 1 __ 2 v.

36. 〈2, −1⟩ 37. 〈−1, 4⟩ 38. 〈−3, −2⟩

For the following exercises, use the vectors shown to sketch u + v, u − v, and 2u.

39.

v u

40.

v

u

41.

v

u

For the following exercises, use the vectors shown to sketch 2u + v.

42.

u

v

43.

uv

For the following exercises, use the vectors shown to sketch u − 3v.

44. u v45.

u

v

For the following exercises, write the vector shown in component form.

46. 47.

SECTION 8.8 section exercises 745

48. Given initial point P 1 = (2, 1) and terminal point P 2 = (−1, 2), write the vector v in terms of i and j, then draw the vector on the graph.

49. Given initial point P 1 = (4, − 1) and terminal point P 2 = (−3, 2), write the vector v in terms of i and j. Draw the points and the vector on the graph.

50. Given initial point P 1 = (3, 3) and terminal point P 2 = (−3, 3), write the vector v in terms of i and j. Draw the points and the vector on the graph.

exTenSIOnS

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

51. | v | = 6, θ = 45 ° 52. | v | = 8, θ = 220° 53. | v | = 2, θ = 300° 54. | v | = 5, θ = 135°

55. A 60-pound box is resting on a ramp that is inclined 12°. Rounding to the nearest tenth,a. Find the magnitude of the normal (perpendicular)

component of the force.b. Find the magnitude of the component of the force

that is parallel to the ramp.

56. A 25-pound box is resting on a ramp that is inclined 8°. Rounding to the nearest tenth,a. Find the magnitude of the normal (perpendicular)

component of the force.b. Find the magnitude of the component of the force

that is parallel to the ramp.

57. Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27° above the horizontal. Round to the nearest hundredth.

58. Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of 127° above the horizontal. Round to the nearest hundredth.

59. Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55° above the horizontal. Round to the nearest hundredth.

60. Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of 8° above the horizontal. Round to the nearest hundredth.

ReAl-WORlD APPlICATIOnS

61. A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?

62. A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?

63. A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?

64. A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?

65. A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction?

66. A woman starts walking from home and walks 6 miles at 40° north of east, then 2 miles at 15° east of south, then 5 miles at 30° south of west. If she walked straight home, how far would she have to walk, and in what direction?

Chapter 13.Trigonometric Equations

343

13-2 Using Factoring to SolveTrigonometric Equations (pages 529–530)

Writing About Mathematics1. No. The method of using factoring to solve a

trigonometric equation depends on themultiplicative property of zero: If ab 5 0, then a 5 0 or b 5 0. Thus, the right side of theequation must equal 0.

2. Yes. 2(sin u)(cos u) 1 sin u 1 2 cos u 1 1 5 0

sin u (2 cos u 1 1) 1 1(2 cos u 1 1) 5 0

(sin u 1 1)(2 cos u 1 1) 5 0

sin u 1 1 5 0 2 cos u 1 1 5 0Developing Skills

3. 30°, 150°, 270° 4. 30°, 150°, 210°, 330°5. 60°, 120°, 240°, 300° 6. 45°, 135°, 225°, 315°7. 60°, 300° 8. 90°, 210°, 270°, 330°9. 45°, 63.4°, 225°, 243.4° 10. 0°, 70.5°, 289.5°

11. 19.5°, 41.8°, 138.2°, 160.5°12. 66.4°, 113.6°, 246.4°, 293.6°13. 63.4°, 99.5°, 243.4°, 279.5°14. 70.5°, 75.5°, 284.5°, 289.5°15. 1.11, 1.25, 4.25, 4.39 16. 0, 1.82, 4.4617. 0, p (3.14), 3.55, 5.87 18. 3.48, 5.9419. 0.25, 0.52, 2.62, 2.89 20. 0.17, 1.11, 3.31, 4.2521. 22. 2.03

23. 290°, 270°

13-3 Using the Quadratic Formula toSolve Trigonometric Equations (page 534)

Writing About Mathematics1. When the discriminant is negative, the solutions

are imaginary numbers.2. When factored, and .The

range of the cosecant function is.

Developing Skills3. 202°, 338° 4. 74°, 125°, 254°, 305°5. 29°, 99°, 261°, 331° 6. 14°, 166°, 246°, 294°7. 111°, 159°, 291°, 339° 8. 46°, 80°, 280°, 314°9. 50°, 157°, 230°, 337° 10. 72°, 144°, 216°, 288°

11. 55°, 125° 12. 39°, 119°, 219°, 299°13. 64°, 140°, 220°, 296° 14. { }15. 16. 0.56, 5.72p

4 , 3p4 , 5p

4 , 7p4

(2`, 21g < f1, `)

csc u 5 12csc u 5 0

308 or p6

13-1 First Degree TrigonometricEquations (pages 524–526)

Writing About Mathematics1. The second equation simplifies to sin x 5 2 and 2

is outside the range of the sine function.2. The second equation simplifies to tan x 5 1.

Since the tangent function is periodic, there arean infinite number of x-values where tan x 5 1.

Developing Skills3. 60°, 300° 4. 150°, 330°5. 90° 6. 90°, 270°7. 135°, 315° 8. 45°, 315°9. 10. 0

11. 12.13. 14.15. 49° 16. 79°17. 71° 18. 24°19. 12° 20. 18°21. 75.5°, 284.5° 22. 16.6°, 163.4°23. 104.0°, 284.0° 24. 131.8°, 228.2°25. 0.17, 2.97 26. 2.09, 4.19 or ,27. 0.38, 3.52 28. 3.02, 6.16Applying Skills29. a.

b. 220 volts c. 2d. (1) u 5 0.93, 5.36

(2) t 5 0.30 s and 1.70 s30. a. x 5 36.9° b. u 5 18.4°31. a. We used the expression for the cotangent

function, which is undefined at .

b. Yes.

.sin Ap2 2 p

2 B 5 0

5 01 5 0 and

cos p2sin p2

cot Ap2) 5

p2

1tan u

E

t1

10

25

215

5

210

15

2O

4p3

2p3

p2 , 3p

2p6 , 5p

6

5p4 , 7p

4p4 , 5p

4

p3 , 2p

3

14580AK05.pgs 3/26/09 12:07 PM Page 343

13-4 Using Substitution to SolveTrigonometric Equations InvolvingMore Than One Function (pages 537–538)

Writing About Mathematics1. Yes. The maximum value of both sin u and cos u

is 1, and they are never equal to 1 for the samevalue of u. Therefore, their sum will always beless than 2.

2. 0° # u # 180°. Sine is positive in the first andsecond quadrants.

Developing Skills3. 30°, 150° 4. 30°, 150°5. 45°, 90°, 225°, 270° 6. 30°, 150°, 270°7. 0°, 60°, 300° 8. 30°, 150°, 210°, 330°9. 45°, 135°, 225°, 315° 10. 90°, 270°

11. 45°, 135°, 225°, 315° 12. 0°, 180°, 210°, 330°13. 30°, 150° 14. 45°, 135°, 225°, 315°Applying Skills15. a. (1) (2)

b. 0° c. 628.96°

13-5 Using Substitution to SolveTrigonometric Equations InvolvingDifferent Angle Measures (pages 540–541)

Writing About Mathematics1. No. Dividing by 2 divides the coefficients, not the

angles.2. No. You must account for the factor cos u. The

solution set also includes the numbers .Developing Skills

3. 30°, 90°, 150°, 270° 4. 0°, 180°, 360°5. 0°, 180°, 360°6. 30°, 90°, 150°, 210°, 270°, 330°7. 30°, 150°, 210°, 330° 8. 60°, 300°9. 30°, 90°, 150° 10. 45°, 135°, 225°, 315°

11. 12. 0, 1.82, p, 4.46, 2p

13. 14.

15. 0, 1.15, 1.99, p, 4.29, 5.13, 2p

16. 1.36, 4.92 17. 0.28, 2.86 18. 0.12, 3.02Applying Skills19. a.

b.c.

d. 49.92 mu 5 arctan 23 5 33.698

tan u 5 23

sin u 5 23 cos u

90 sin u 5 60 sin (908 2 u)d 5 60 sin (908 2 u)d 5 90 sin u

0.17, p2 , 2.97, 3p20.34, p2 , 2.80

p5 , 3p

5 , 7p5 , 9p

5

Up2 , 3p

2 V

2!2 2 2 in.2!3 2 2 in.

344

20. a.b.

, reject

Review Exercises (page 543)1. 120°, 240° 2. 240°, 300°3. 60°, 300° 4. 60°, 180°, 300°5. 0°, 180, 360° 6. 60°, 120°, 240°, 300°7. 45°, 135°, 225°, 315° 8. 45°, 135°, 225°, 315°9. 22.5°, 202.5°

10. 30°, 90°, 150°, 210°, 270°, 330°11. 3.43, 5.99 12. { }13. , 3.39, 6.03 14. 0, 2.30, 3.9815. 1.34, 2.91, 4.48, 6.05 16. 1.20, 1.43, 4.85, 5.0817. 0, 0.62, 2.53, p, 3.76, 5.6718. , 3.39, , 6.03 19. 1.33, 4.47

20. 1.23, , 5.05 21. 1.11, 1.77, 4.25, 4.91

22.

23. The left side of the equation is equal to zero onlyat values of u for which both the tangent andsecant functions are undefined.

However, tangent is undefined at .

24. a.

b.

c.

d.

e.

Exploration (page 544)(1) a. A 5 4 tan u b. 0.12(2) a. b. 0.24(3) a. b.

(4) a. b.

(5) a.

Total area 5 p2 2 1

2 sin u cos u

Area of semicircle 5 p2

212 sin u cos u5 12 sin (p 2 u) cos (p 2 u) 5

5 12bh

Area of triangle

p2A 5 1

2 sin u

p4A 5 1

2 sin 2u 5 sin u cos uA 5 2 tan u

m/A 5 248, m/B 5 488, m/C 5 1088

u 5 arctan !55 5 24.018

tan u 5 CDAD 5

!55

CD 5!55 AD

tan 2u 5 CDDB

tan u 5 CDAD

u 5 p2

u 5 p2

sin u 5 1tan u 2 sec u 5 sin u

cos u 2 1cos u 5 0

p3 , 5p

3

2p3 , 4p

3

3p2

p2

p2

u 5 arccos 78 5 28.968

cos u 5 1.752 5 7

8

sin u 5 0sin u (2 cos u 2 1.75) 5 02 sin u cos u 5 1.75 sin u

sin 2u 5 1.75 sin u sin u 5

yx, sin 2u 5

1.75yx 5

7y4x

14580AK05.pgs 3/26/09 12:07 PM Page 344

b. No possible value for u. Since the area of thesemicircle circle is square units, theminimum area of the shaded region is square units.

(6) a.b.

Cumulative Review (pages 545–546)Part I

1. 3 2. 1 3. 44. 1 5. 2 6. 37. 4 8. 4 9. 2

10. 1Part II11.

12.

f21 (x) 5 x 1 2

4

x 1 24 5 y

x 1 2 5 4yx 5 4y 2 2

y

xO

121

p12

A 5 2 sin u cos u 5 sin 2u

p2 . 1

p2 < 1.57

345

Part III

13. ,

14. Center 5

Radius 5

(x 2 1)2 1 (y 2 2)2 5 72Part IV

15.

16.

x 5 32 6 1

2i

x 56 6 !36 2 4(2)(5)

4

0 5 2x2 2 6x 1 5 5 0

22 5 5 2 6xx2

22 5 5x2 2 6

x

6x 2 2 5 5

x2

sec u 5 23!2

4

cot u 5 22!2tan u 5 2!24

cos u 5 22!2

3sin u 5 13

5 6!2

5 !36 1 36"(22 2 4)2 1 (5 1 1)2

A22 1 42 , 5 2 1

2 B 5 (1, 2)

2!3 81, 18, 18!3 9, 18!3 812, 2!3 9

Chapter 14.Trigonometric Applications

14-1 Similar Triangles (pages 551–552)Writing About Mathematics

1. Yes, since , u can be found byusing arctan.

2. Quadrant III. Both cosine and sine are negativewhen evaluated.

Developing Skills

3. 4.

5. (0, 6) 6.

7. 8. (20.5, 0)

9. 10.

11. (0, 212) 12. (21, 21)

13. 14. A1, 2!3BQ!32 , 23

2R

A2252 !3, 225

2 BA292!3, 92 B

A2152 !2, 15

2 !2 BA24, 4!3BA!3, 1BA2!2, 2!2B

tan u 5 28.4825.30 5 1.6

15. a. 10 b. 53°16. a. 13 b. 113°17. a. 7 b. 90°18. a. 15 b. 323°19. a. 15 b. 0°20. a. 14.42 b. 236°21. a. 25 b. 16°22. a. 11.66 b. 301°23. a. 11.31 b. 135°

24. a. R(5, 0), b. sq units

25. a. R(12, 0), S(0, 8) b. 48 sq units

26. a. R(8, 0), b. sq units

27. a. R(20, 0), S(10, 10) b. 100 sq units28. a. R(9, 0), b. sq units29. a. R(7, 0), b. 28 sq unitsSA8!3, 8B

20.25!3SA4.5, 4.5!3B16!2SQ24!2, 4!2R

15!34SQ1.5, 3!3

2 R

14580AK05.pgs 3/26/09 12:07 PM Page 345

TRY IT ANSWERSB-8

Section 8.71.

y

(−12, −2) (8, −2)

(−2, 3)

(−2, −2)x

x = −2

y = 8

(x + 2)2 = −20(y − 3)

2. y

x0.25

0.25

0.5 0.75−0.75

−0.25

−0.25

0.5

−0.5

−0.5

3.

2

y

–1–1–2–2

–3

–3

–4

–4

–5

–5

1

3

3

2 4

4

5

5

1

The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.

Section 8.8 1.

2

x

y

1– –1

1

345

1 2 3 4 5 6

⟨3, 5⟩

2. 3u = ⟨15, 12⟩

3. u = 8i − 11j

4. v = √—

34 cos(59°)i + √—

34 sin(59°)j

Magnitude = √—

34

θ = tan−1 5 _ 3

= 59.04°

Chapter 9

Section 9.11. Not a solution2.

Section 9.21. (1, −1, 1) 2. No solution 3. Infinite number of solutions of the form (x, 4x − 11, −5x + 18)

Section 9.3

1. − 1 _ 2

, 1 _ 2

and (2, 8) 2. (−1, 3) 3. {(1, 3),(1, −3),(−1, 3),(−1, −3)}

4.

4

y

–2–2–4–4

–6

–6

–8–10

2

6

6

4 8

8

10

10

2

(4, 2)(−2, −2) (10, 2)

(0, 2)(8, 2)

(4, 2 + 2 )

(4, 2 − 2 )

Section 9.4

1. 3 _ x − 3

− 2 _ x − 2

2. 6 _ x − 1

− 5 _ (x − 1)2

3. 3 _ x − 1

+ 2x − 4 _ x2 + 1

4. x − 2 _ x2 − 2x + 3

+ 2x + 1 __ (x2 − 2x + 3)2

Section 9.5

1. A + B = 2

6

1 0 1

−3

+ 3

−2

1 5 −4

3

= 2 + 3

6 + (−2)

1 + 1 0 + 5 1 + (−4)

−3 + 3

= 5

4

2 5 −3

  0

2. −2B = −8 −2 −6 −4

Section 9.6

1. 4 −3 3 2 11 4 2. x − y + z = 5 3. (2, 1) 2x − y + 3z = 1 y + z = −9

4. 1

− 5 __ 2

5 __ 2

0 1 5 0

0

1

17 __ 2 9

2 5. (1, 1, 1)

6. $150,000 at 7%, $750,000 at 8%, $600,000 at 10%

Section 9.7

1. AB = 1 4 −1 −3 −3 −4 1 1 = 1(−3) + 4(1) 1(−4) + 4(1) −1(−3) + −3(1)

  −1(−4) + −3(1)

= 1 0 0 1

BA = −3 −4 1 1 1 4 −1 −3 = −3(1) + −4(−1)   −3(4) + −4(−3) 1(1) + 1(−1)   1(4) + 1(−3)

= 1 0 0 1

2. A−1 = 3 __ 5

1 __ 5

− 2 __ 5

1 __ 5

3. A−1 = 1 1

2

2 4 −3 3

6

−5

4. X = 4 38

58

Section 9.81. (3, −7) 2. −10 3. −2, 3 _ 5 , 12 _ 5

Chapter 10

Section 10.1

1. x2 + y2

_ 16

= 1 2. (x − 1)2

_ 16

+ (y − 3)2

_ 4 = 1

3. Center: (0, 0); Vertices: (±6, 0); Co-vertices: (0, ±2); Foci: (±4 √

— 2 , 0)

4. Standard form: x2 _

16 +

y2

_ 49

= 1; Center: (0, 0); Vertices: (0, ±7); Co-vertices: (±4, 0) Foci: (0, ± √

— 33 )

3. (−2, −5)4. (−6, −2)5. (10, −4)6. No solution. It is an inconsistent system.7. The system is dependent so there are infinite solutions of the form (x, 2x + 5). 8. 700 children, 950 adults

The solution to the system is the ordered pair(−5, 3).

2

y

–1–1–2–2

–3

–3

–4

–4

–5

–5–6–7

1

3

3

2 4

4

5

567

1

(0, 0) (4, 0)(−4, 0)

(0, 7)

(0, −7)−

2

y

–1–1–2–2

–3

–3

–4

–4

–5

1

3

3

2 4

4

51

(0, 0) (6, 0)(−6, 0)

(0, 2)

(0, −2)

2

y

–1–1–2–2

–3

–3

–4

–4

–5

–5–6–7

1

3

3

2 4

4

5

567

1

(0, 0) (4, 0)(−4, 0)

(0, 7)

(0, −7)−

ODD ANSWERS C-33

Section 8.81. Lowercase, bold letter, usually u, v, w 3. They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1. 5. The first number always represents the coefficient of the i, and the second represents the j. 7. ⟨7, − 5⟩ 9. Not equal 11. Equal 13. Equal 15. −7i − 3j 17. −6i − 2j19. u + v = ⟨−5, 5⟩, u − v = ⟨−1, 3⟩, 2u − 3v = ⟨0, 5⟩

21. −10i − 4j 23. − 2 √—

29 _ 29 i + 5 √—

29 _ 29 j

25. − 2 √—

229 _ 229 i + 15 √—

229 _ 229 j 27. − 7 √— 2 _ 10 i + √

— 2 _ 10 j

29. |v| = 7.810, θ = 39.806° 31. |v| = 7.211, θ = 236.310°33. −6 35. −1237. yy

x

y

v

x

3v

x

v12

39.

2uu + v

u − v

41.

u + v 2u

u − v

43. 45.

47. ⟨4, 1⟩ 49. v = −7i + 3j51. 3 √

— 2 i + 3 √

— 2 j

53. i − √— 3 j

55. a. 58.7; b. 12.557. x = 7.13 pounds, y = 3.63 pounds59. x = 2.87 pounds, y = 4.10 pounds61. 4.635 miles, 17.764° N of E63. 17 miles, 10.071 miles65. Distance: 2.868, Direction: 86.474° North of West, or 3.526° West of North 67. 4.924°, 659 km/hr 69. 4.424° 71. (0.081, 8.602)73. 21.801°, relative to the car’s forward direction75. Parallel: 16.28, perpendicular: 47.28 pounds77. 19.35 pounds, 51.65° from the horizontal79. 5.1583 pounds, 75.8° from the horizontal

Chapter 8 Review exercises1. Not possible 3. C = 120°, a = 23.1, c = 34.15. Distance of the plane from point A: 2.2 km, elevation of the plane: 1.6 km 7. B = 71.0°, C = 55.0°, a = 12.8 9. 40.6 km11. 13. (0, 2)

15. (9.8489, 203.96°) 17. r = 819. x 2 + y 2 = 7x

21. y = − x

123

x

y

−2−1

−1 1 2 3 4 5 6 7−2−3−4−5−6−7

−4−5−6−7

−3

4567

23. Symmetric with respect to the line θ = π __ 2

25.

1 2 3 4 5 6

(θ from 0 to 2π) 27.

1 2 3 4 5 6

(θ from 0 to 2π)

29. 5 31. cis − π __ 3 33. 2.3 + 1.9i 35. 60cis π __ 2 37. 3cis 4π ___ 3 39. 25cis 3π ___ 2 41. 5cis 3π ___ 4 , 5cis 7π ___ 4 43.

123

Real

Imaginary

−2−1

−1 1 2 3 4 5 6−2−3−4−5−6

−4−5−6

−3

456

45. x 2 + 1 __ 2 y = 1

47. x(t) = −2 + 6t

y(t) = 3 + 4t

49. y = −2x 5

x

y

−20

20

−40

−60

−80

−100

0.5 1.0 1.5 2.0

(t from −1 to 1)

51. a.

x(t) = (80 cos (40°))t

y(t) = − 16t 2 + (80 sin (40°))t + 4

b. The ball is 14 feet high and 184 feet from where it was launched.c. 3.3 seconds53. Not equal 55. 4i

57. − 3 √—

10 _ 10 i, − √—

10 _ 10 j

59. Magnitude: 3 √— 2 , Direction: 225°

61. 16

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