slide 8- 1 copyright © 2006 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Trigonometric Functions

Chapter 5


5.1 Trigonometric Functions of

Acute Angles Determine the six trigonometric ratios for a given acute

angle of a right triangle.

Determine the trigonometric function values of 30°, 45°, and 60°.

Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle.

Given the function values of an acute angle, find the function values of its complement.


Trigonometric Ratios

The figure illustrates how a right triangle is labeled with reference to a given acute angle, .

The lengths of the sides of the triangle are used to define the six trigonometric ratios:sine (sin) cosecant (csc)cosine (cos) secant (sec)tangent (tan) cotangent (cot)


Sine and Cosine

The sine of is the length of the side opposite divided by the length of the hypotenuse:

The cosine of is the length of the side adjacent to divided by the length of the hypotenuse.

side opposite sin

hypotenuse

side adjacent to cos

hypotenuse

Side Adjacent

to

Side Opposite

Hypotenuse


Trigonometric Function Values of an Acute Angle


Example

Use the triangle shown to calculate the six trigonometric function values of .

Solution:

24sin

25

7cos

25

24tan

7

opp

hyp

adj

hyp

opp

adj

25csc

24

25sec

7

7cot

24

hyp

opp

hyp

adj

adj

opp

7

24 25


Reciprocal Functions

Reciprocal Relationships

1csc

sin

1sec

cos

1cot

tan


Pythagorean Theorem

The Pythagorean theorem may be used to find a missing side of a right triangle.

This procedure can be combined with the reciprocal relationships to find the six trigonometric function values.

2

5 h

2 2 2

2

2

2 5

4 25

29

29

h

h

h

h


Example

If the find the other five trigonometric function values of .

Solution: Find the length of the hypotenuse.

5tan ,

2

2

529

5 5 29sin

2929

2 2 29cos

29295

tan2

29csc

5

29sec

22

cot5


Function Values of 30 and 60

When the ratio of the opposite side to the hypotenuse is ½, must have a measure of 30.

Using the Pythagorean theorem the missing side is The missing angle must have a measure of 60.

3.

1sin

2

30

21


Function Values of 30 and 60

1sin30

2

3cos30

2

1 3tan30

33

3sin 60

21

cos602

tan 60 3

21

30

60

3


Function Values of 45

The legs of this triangle must be equal, since they are opposite congruent angles.

The hypotenuse is found by:

2 2 2

2

2

1 1

1 1

2

2

h

h

h

h

1

1h

45

45


Function Values of 45 continued

1

1

45

45

2

1 2sin 45

22

1 2cos45

221

tan 45 11


Summary of Function Values


Example

As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.)


Example continued

Solution: We begin with a drawing of the situation. We know the measure of an acute angle and the length of its adjacent side.


Example continued

Since we want to determine the length of the opposite side, we can use the tangent ratio, or the cotangent ratio.

The balloon is approximately 0.7 mi, or 3696 ft, high.

tan301.2

1.2 tan30

31.2

3

0.7

opp h

adj

h

h

h


Cofunctions and Complements

The trigonometric function values for pairs of angles that are complements have a special relationship. They are called cofunctions.

sin cos(90 )

tan cot(90 )

sec csc(90 )

cos sin(90 )

cot tan(90 )

csc sec(90 )


Example

Given that sin 40 0.6428, cos 40 0.7660, and tan 40 0.8391, find the six trigonometric function values of 50.

1csc40 1.5557

sin 401

sec40 1.3055cos40

1cot 40 1.1918

tan 40

sin50 cos40 0.7660

tan50 cot 40 1.1918

sec50 csc40 1.5557

cos50 sin 40 0.6428

cot50 tan 40 0.8391

csc50 sec40 1.3055


5.2 Applications of Right Triangles

Solve right triangles.

Solve applied problems involving right triangles and trigonometric functions.


Solving Right Triangles

To solve a right triangle means to find the lengths of all sides and the measures of all angles. This can be done using right triangle trigonometry.


Example

In , find a, b, and B.

Solution:

ABC

16.5 a

bA

B

C42

sin 42

16.5

16.5sin 42

11.0

a

a

a

cos4216.5

16.5cos42

12.3

b

b

b

B = 90 42 = 48


Definitions

Angle of elevation: angle between the horizontal and a line of sight above the horizontal.

Angle of depression: angle between the horizontal and a line of sight below the horizontal.


Example

To determine the height of a tree, a forester walks 100 feet from the base of the tree. From this point, he measures the angle of elevation to the top of the tree to be 47. What is the height of the tree?

tan 47100

100 tan 47

107.2 ft

h

h

h

100 ft

h

47


Bearing

Bearing is a method of giving directions. It involves acute angle measurements with reference to a north-south line.


Example

An airplane leaves the airport flying at a bearing of N32W for 200 miles and lands. How far west of its starting point is the plane?

The airplane is approximately 106 miles west of its starting point.

sin32200

200sin32

106

w

w

w

w

200

32


5.3 Trigonometric Functions of

Any Angle Find angles that are coterminal with a given angle and find

the complement and the supplement of a given angle.

Determine the six trigonometric function values for any angle in standard position when the coordinates of a point on the terminal side are given.

Find the function values for any angle whose terminal side lies on an axis.

Find the function values for an angle whose terminal side makes an angle of 30°, 45°, or 60° with the x-axis.

Use a calculator to find function values and angles.


Angle in Standard Position

An angle formed by it’s initial side along the positive x-axis, with it’s vertex at the origin, and it’s terminal side placed at the end of the rotation.


Coterminal Angles

Two or more angles that have the same terminal side. For example, angles of measure 60 and 420 are

coterminal because they share the same terminal side.

Example: Find two positive and two negative angles that are coterminal with 30.

390, 750, 330, and 690 are coterminal with 30.

30 360 390

30 2(360 ) 750

30 360 330

30 2(360 ) 690


Trigonometric Functions of Any Angle


Example

Find the six trigonometric function values for the angle shown:

Solution: First, determine r.

2

4r

(2,4)

2 22 4

20

4 5 4 5

2 5

r

r

r

r


Example continued

The six trigonometric functions values are:

4 2 5sin

52 5

2 5cos

52 54

tan 22

y

r

x

r

y

x

2 5 5csc

4 2

2 5sec 5

22 1

cot4 2

r

y

r

xx

y


Another Example

Given that and is in the first quadrant, find

the other function values.

Solution: Sketch and label the angle. Find any missing sides.

1tan ,

2

2 22 1

4 1

5

r

r

r

1

2

r

(2,1)

1 5sin

55

2 2 5cos

551

tan2

5csc 5

1

5sec

22

cot 21


Reference Angle

The reference angle for an angle is the acute angle formed by the terminal side of the angle and the x-axis.

The reference angle can be used when trying to find the trigonometric function values for angles that cover more than one quadrant. (ex. 210)


Example

Find the sine, cosine, and tangent function values for 210.

Solution: Draw the angle.

Note that there is a 30 angle in the third quadrant. Label the sides of the triangle with 1, and 2 as

shown. 3,

1 2

210

303


Example continued

Notice that both the sine and cosine are negative because the angle measuring 210 is in the third quadrant.

1sin 210

2

3cos210

21

tan 2102

1 2

210

303


5.4 Radians, Arc Length, and

Angular Speed Find points on the unit circle determined by real

numbers.

Convert between radian and degree measure; find coterminal, complementary, and supplementary angles.

Find the length of an arc of a circle; find the measure of a central angle of a circle.

Convert between linear speed and angular speed.


Radian Measure

An angle measures 1 radian when the angle intercepts an arc on a circle equal to the radius of the circle.

1 radian is approximately 57.3.


Converting between Degree and Radian Measure

To convert from degree to radian measure, multiply by

To convert from radian to degree measure, multiply by

radians 1801.

180 radians

radians.

180

180.

radians


Example

Convert each of the following to either radians or degrees.

a) 150 b) 75

radians

c) radians d) 3 radians7

4

150 5150

180 180 6

75 5

75180 180 12

7 180 1260315

4 4

180 540

3 171.9


Radian Measure


Example

Find the length of an arc of a circle of radius 10 cm associated with an angle of radians.

4

ors

s rr

10 510

4 4 27.85 cm

s r

s

s


Definitions

Linear Speed: the distance traveled per unit of time,

where s is the distance and t is the time.

Angular Speed: the amount of rotation per unit of time, where is the angle of rotation and t is the time.

sv

t

t


Linear Speed in Terms of Angular Speed

The linear speed v of a point a distance r from the center of rotation is given by v = r, where is the angular speed in radians per unit of time.


Example

Find the angle of revolution of a point on a circle of diameter 30 in. if the point moves 4 in. per second for 11 seconds.

Since t = 11, must be determined before we can solve for .

, .v

v rr

4in./ sec0.26 per second

15in.4 44

1115 15

2.93

t

The angle of revolution of the point is approximately 3 radians.


5.5Circular Functions:

Graphs and Properties Given the coordinates of a point on the unit circle, find

its reflections across the x-axis, the y-axis, and the origin.

Determine the six trigonometric function values for a real number when the coordinates of the point on the unit circle determined by that real number are given.

Find function values for any real number using a calculator.

Graph the six circular functions and state their properties.


Basic Circular Functions


Example

Find each of the following function values.

a) b)

c) d)

Solutions:

a) The coordinates of the point

determined by are

2sin

3

cos

4

3tan

4

5sec

6

2

3

1 3,

2 2

2 3sin

3 2y

2

3

1 3,

2 2


Example continued

b) The coordinates of the point determined by

are

c) The coordinates of the point determined by

are

d) The coordinates of the point determined by

are

4

2cos

4 2x

3

4

5

6

3 2 2tan 1

4 2 2

y

x

5 1 1 2 3sec

6 33 2x

2 2,

2 2

2 2,

2 2

3 1,

2 2


Graphs of the Sine and Cosine Functions

1. Make a table of values.

2. Plot the points.

3. Connect the points with a smooth curve.


Example

1. Make a table of values.


Example continued

2. Plot the points.

3. Connect the

points with

a smooth curve.


Domain and Range of Sine and Cosine Functions

The domain of the sine and cosine functions is (, ).

The range of the sine and cosine functions is [1, 1].

Periodic FunctionA function f is said to be periodic if there exists a positive constant p such that f(s + p) = f(s) for all s in the domain of f. The smallest such positive number p is called the period of the function.


Amplitude

The amplitude of a periodic function is defined as one half of the distance between its maximum and minimum function values.

The amplitude is always positive.

The amplitude of y = sin x and y = cos x is 1.


Graph of y = tan s


Graph of y = cot s


Graph of y = csc s


Graph of y = sec s


5.6Graphs of Transformed Sine

and Cosine Functions Graph transformations of y = sin x and y = cos x in the

form y = A sin (Bx – C) + D and y = A cos (Bx – C) + D and determine the amplitude, the period, and the phase shift.

Graph sums of functions.


Graphs of Transformed Sine and Cosine Functions: Vertical Translation

y = sin x + D and y = cos x + D

The constant D translates the graphs D units up if D > 0 or |D| units down if D < 0.

Example:

Sketch a graph of

y = sin x 2.


Graphs of Transformed Sine and Cosine Functions: Amplitude

y = A sin x and y = A cos x

If |A| > 1, then there will be a vertical stretching by a factor of |A|.

If |A| < 1, then there will be a vertical shrinking by a factor of |A|.

If A < 0, the graph is also reflected across the x-axis.


Example

Sketch a graph of y = 3 sin x.

The sine graph (y = sin x) is stretched vertically by a factor of 3.


Graphs of Transformed Sine and Cosine Functions: Period

y = sin Bx and y = cos Bx

If |B| < 1, then there will be a horizontal stretching.

If |B| > 1, then there will be a horizontal shrinking.

If B < 0, the graph is also reflected across the y-axis.

The period will be 2

.B


Example

Sketch a graph of y = sin 2x.

The sine graph (y = sin x) is shrunk horizontally.

The period is 2

2


Graphs of Transformed Sine and Cosine Functions: Horizontal Translation or Phase Shift

y = sin (x C) and y = cos (x C)

The constant C translates the graph horizontally |C| units to the right if C > 0 and |C| units to the left if C < 0.


Example

Sketch the graph of y = sin (x + ).

The sine graph (y = sin x) is translated units to the left.


Combined Transformations

y = A sin (Bx C) + D and y = A cos (Bx C) + D

The amplitude is |A|.

The period is .

The graph is translated vertically D units.

The graph is translated horizontally C units.

2

B


Example

Find the vertical shift, amplitude, period, and phase shift for the following function:y = 2 sin (4x 2) 3.

Solution: Write the function in standard form.

|A| = |2| = 2 means the amplitude is 2

B = 4 means the period is

C/B = means the phaseshift is units to the right.

D = 3 means the vertical shift is 3 units down.

2sin(4 2 ) 3

22sin 4 ( 3)

4

2sin 4 ( 3)2

y x

y x

y x

2

4 2

2

2


Example continued

Sketch Amplitude = 2 Vertical shift = 3 down Phase shift = right Period =

First, sketch y = sin 4x.

2sin 4 ( 3)2

y x

2

2


Example continued

Second, sketch

y = 2 sin 4x.

Third, sketch

.

2sin 42

y x


Example continued

Finally, sketch

.2sin 4 ( 3)2

y x

slide 8- 1 copyright © 2006 pearson education, inc. publishing as pearson addison-wesley

Documents