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Name: ______________ 

MAT Final Review Packet All assignments are due the day after they are assigned.

Date HW

In Class Homework Assignments

M 6/9 38

Notes #38: Review for Extra Exam

Final Review (#2-42 even)

T 6/10 39

Extra Exam Trig Review Worksheet (Pg. 19-20 in this Packet)

W 6/11 40

Notes #40: Review for Final

More Trig Review Worksheet (Pg. 17-18 in this Packet)

F 6/13 43

Final

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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NOTES #38: REVIEW LESSON 1 – USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES

FUNDAMENTAL TRIGONOMETRIC INDENTITIES

Reciprocal Identities

1sin

csc

1

cscsin

1cos

sec

1

seccos

1tan

cot

1

cottan

Quotient Identities

sintan

cos

cos

cotsin

Pythagorean Identities

2 2sin cos 1 2 21 tan sec 2 21 cot csc

Cofunction Identities

sin 90 cos

sin cos2

tan 90 cot

tan cot2

sec 90 csc

sec csc2

cos 90 sin

cos sin2

cot 90 tan

cot tan2

csc 90 sec

csc sec2

Even/Odd Identities

Even Functions cos( ) cost t sec( ) sect t

Odd Functions sin( ) sint t csc( ) csct t tan( ) tant t cot( ) cott t

3

1. Practice Problem 1:

Given 3

sin2

x and

1cos

2x . Find:

tan x csc x

cot x

sec x

1. Practice Problem 2:

Given 3

cos2 5

x

and 4

cos5

x . Find:

sin x csc x

tan x sec x

sin2

x

cot x

Practice Problems: Simplify the following trig expressions 3. sin tan cos

4. 2

1

tan 1x

5. cos 1 sin

1 sin cos

x x

x x

6. 2sin cosx x

4

Practice Problems: Factor each expression. 7. 24 tan tan 3

8. 3sin 8

Practice Problem 9: Rewrite 1

1 sin x so that it is not in fractional form.

Practice Problem 10: Use the trig substitution 3cos , 02

x to express

29 x as a trig function of .

Practice Problem 11: Use the trig substitution to write the algebraic equation as a

trig function of where 2 2

. Then find sin and cos .

22 2 16 4 , 2cosx x

5

REVIEW LESSON 2 – VERIFYING TRIGONOMETRIC IDENTITIES Review Conditional Equation

Only true for some of the value in its domain. sin 0x x n where n is an integer

Identity True for all real values in its domain.

2 2sin 1 cosx x True for all real numbers x.

Guidelines for Verifying Trig Identities

1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.

2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.

3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents.

4. If the preceding guidelines do not help, try converting all terms to sines and cosines.

5. Always try something. Even paths that lead to dead ends give you insights.

Practice Problems: Verify the identities

1. 2csc

csc seccot

2. 1 sin cos

2seccos 1 sin

6

3. 2 4 2 4sin sin cos cos

4. tan cot sec cscx x x x

5. 2cot 1 sin

1 csc sin

6. cos

sec tan1 sin

yy y

y

7

7. 1 cos 1 cos

1 cos sin

8. 2 2sec 1 cot2

x x

8

REVIEW LESSON 3 – SOLVING TRIGONOMETRIC EQUATIONS Practice Problems: Solve the equations. 1. 2cos 1 0x

2. sin 2 sinx x

3. 23sec 4 0x

4. 2cot cos 2cotx x x

5. Solve the equation by squaring and converting to quadratic type. Check your solutions. cos 1 sinx x

9

Practice Problems: Find all solutions of the equation in the interval 0, 2 .

6. 2sec sec 2x x

7. cos sin tan 2x x x

8. 212sin 13sin 3 0x x

9. 2tan 6 tan 5 0x x

10. tan 3 1x

11. 23tan 2 1 0x

10

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NOTES #40: REVIEW LESSON 4 – SUM AND DIFFERENCE FORMULAS Sum and Difference Formulas

sin sin cos cos sinu v u v u v

sin sin cos cos sinu v u v u v

cos cos cos sin sinu v u v u v

cos cos cos sin sinu v u v u v

tan tantan

1 tan tan

u vu v

u v

tan tantan

1 tan tan

u vu v

u v

Practice Problem 1: Find the exact value. a. sin105

b. cos105 c. tan105

Practice Problem 2: Find the exact value.

a. 11

sin12

b. 11

cos12

c.

11tan

12

12

Practice Problem 3: Write the expression as the sine, cosine, or tangent of an angle. a. cos25 cos15 sin 25 sin15

b. tan 2 tan

1 tan 2 tan

x x

x x

Practice Problem 4: Find the exact value of the expression. a. cos15 cos60 sin15 sin 60

b. tan 25 tan110

1 tan 25 tan110

Practice Problems: Verify the identity

5. 1 tan

tan4 1 tan

6. sin 3 sinx x

13

Practice Problem 7: Find the exact value of the trig function given that 5

sin13

u

and 3

cos5

v . (Both u and v are in Quad II)

a. sec v u

b. cot u v

Practice Problems: Find all solutions of the equation in the interval 0, 2 .

8. sin sin 13 3

x x

9. tan cos 02

x x

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REVIEW LESSON 5 – MULTIPLE-ANGLE Double Angle Formulas

sin 2 2sin cosu u u

2 2 2 2cos 2 cos sin 2cos 1 1 2sinu u u u u

2

2 tantan 2

1 tan

uu

u

Practice Problems: Find all solutions 1. sin 4 2sin 2x x

2. 2sin 2 cos2 1x x

Practice Problem 2: Use a double-angle formula to rewrite the expression: cos sin cos sinx x x x

Practice Problem 3: Express sin3x in terms of sin x .

15

15

8

Practice Problem 4: Use the following to find sin 2 , cos 2 , tan 2 .and

Given: 3

cot 4; 22

.

a. sin 2

b. cos2 c. tan 2

Half-Angle Formulas

1 cossin

2 2

u u

1 coscos

2 2

u u

1 costan

2 sinsin

1 cos

u u

uu

u

Practice Problem 5: Find the exact value of the trig functions

a. cos2

b. csc2

16

Practice Problems: Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.

1. 3

8

sin2

cos2

tan2

 

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HW #40: More Trig Review for Final Exam: Analytic Trigonometry

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More Trig Review for Final Exam: Analytic Trigonometry

 

 

 

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HW #39: Trig Review-Final Exam

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