Slide 5.3- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 5.3- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Trigonometric Functions of Any Angle

Learn and use the definitions of the trigonometric functions of any angle.Learn and use the signs of the trigonometric functions.Learn to find and use a reference angle.Learn to find the area of an SAS triangle.Learn and use the unit circle definitions of the trigonometric functions.Learn and use some basic trigonometric identities.

SECTION 5.3

1

2

3

4

5

6


DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE


DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

sin y

r

cos x

r

tan y

x, x 0

csc r

y, y 0

sec r

x, x 0

cot x

y, y 0

Let P(x, y) be any point on the terminal ray of an angle in standard position (other than the

r x2 y2 .origin), and let Then r > 0, and:


EXAMPLE 1 Finding Trigonometric Function Values

Suppose that is an angle whose terminal side contains the point P(–1, 3). Find the exact values of the six trigonometric functions of .

Solution

r2 x2 y2

1 2 32

10


EXAMPLE 1 Finding Trigonometric Function Values

Solution continued

sin y

r

3

10

3 10

10

cos x

r

1

10

10

10

tan y

x

3

1 3

csc r

y

10

3

10

3

sec r

x

10

1 10

cot x

y

1

3

1

3

Now, with x 1, y 3 and r 10 we have


TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES

These equations hold for any integer n.

in degrees

sin sin n360º

cos cos n360º

in radians

sin sin 2n

cos cos 2n


TRIGONOMETRIC FUNCTION VALUES OF QUADRANTAL ANGLES

00º 0 1 0 und. 1 und.

deg

sin cos tan csc sec tanradians

180º 0 1 0 und. 1 und.

2360º 0 1 0 und. 1 und.

32

270º 1 0 und. 0 und. 1

2

90º 1 0 und. 0 und. 1


SIGNS OF TRIGONOMETRIC FUNCTIONS


DEFINITION OF A REFERENCE ANGLE

Let be an angle in standard position that is not a quadrantal angle. The reference angle for is the positive acute angle ´(“theta prime”) formed by the terminal side of and the positive or negative x-axis.






EXAMPLE 4 Evaluating Trigonometric Functions

Given that tan 3

2, and cos 0, find the

exact value of sin and sec.

Solution

tan º 0 and cos 0, lies in Quandrant III

x and y are both negative

tan y

x

3

2

3

2

r x2 y2 2 2 3 2 4 9 13


EXAMPLE 4 Evaluating Trigonometric Functions

Solution continued

With x 2, y 3, and r 13 we can find

sin and sec.

sin y

r

3

13

3 13

13

sec r

x

13

2

13

2


PROCEDURE FOR USING REFERENCE ANGLES TO FIND TRIGONOMETRIC

FUNCTION VALUESStep 1 If the degree measure of is greater

than 360º, then find a coterminal angle for with degree measure between 0º and 360º. Otherwise, use in Step 2.

Step 2 Find the reference angle ´ for the angle resulting in Step 1. Write the trigonometric function of the acute angle, ´.


PROCEDURE FOR USING REFERENCE ANGLES TO FIND TRIGONOMETRIC

FUNCTION VALUESStep 3 The sign of a trigonometric function of

depends on the quadrant in which lies. Use the signs of the trigonometric functions to determine when to change the sign of the associated value for ´. (Since ´ is an acute angle, all its function values are positive.)


EXAMPLE 6Using the Reference Angle to Find Values of the Trigonometric Function

a. tan 330º

360º 330º 30º

tan tan 30º 3

3

b. sec59

6

Find the exact value of each expression.

Solution

Step 1 0º < 330º < 360º, find its reference angle

Step 2 330º is in Q IV, its reference angle ´ is



tan 330º tan 30º3

3

Solution continued

Step 3 In Q IV, tan is negative, so

596

11 48

6

116

8b. Step 1

116

is between 0 and 2π coterminal with59

6



sec59

6sec

116

sec6

2 3

3

116

Solution continued

Step 3 In Q IV, sec > 0, so

Step 2

2 11

6

6

sec sec6

2 3

3

is in Q IV, its reference angle ´ is


AREA OF A TRIANGLE

In any triangle, if is the included angle between sides b and c, the area K of the triangle is given by

K 1

2bcsin


EXAMPLE 8 Finding a Triangular Area Determined by Cellular Telephone Towers

Three cell towers are set up on three mountain peaks. Suppose the lines of sight from tower A to towers B and C form an angle of 120º, and the distances between tower A and towers B and C are 3.6 miles and 4.2 miles, respectively. Find the area of the triangle having these three towers as vertices.


EXAMPLE 7 Measuring the Height of Mount Kilimanjaro

SolutionArea of the triangle with angle = 120º included between sides of lengths b = 3.6 and c = 4.2 is

A 1

2bcsin

1

23.6 4.2 sin120º

1

23.6 4.2 sin 60º

1

23.6 4.2 3

26.55 square miles

Finding a Triangular Area Determined by Cellular Telephone Towers


UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS


UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS

Let t be any real number and let P(x, y) be the point on the unit circle associated with t. Then

sin t y

cos t x

tan t y

x, x 0

csc t 1

y, y 0

cot t x

y, y 0

sec t 1

x, x 0


BASIC TRIGONOMETRIC IDENTITIES

Quotient Identities

tan t sin t

cos t cos2 t sin2 t 1

csc t 1

sin t

cot t cos t

sin t

sec t 1

cos t

1 tan2 t sec2 t

1 cot2 t csc2 t

Reciprocal Identities

Pythagorean Identities


EXAMPLE 9Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities

a. Given sin t 1

3 and cos t 0, find cos t and tan t.

b. Given sec t 2 and tan t 0, find tan t.

cos2 t sin2 t 1

cos2 t 1

3

2

1

a. Use Pythagorean identity involving sin t.

Solution

cos2 t 1 1

9



Solutioncos2 t

8

9

cos t 8

9

2 2

3

cos t 2 2

3cos t 0 is given

tan t sin t

cos t

13

2 2

3

1

2 2

2

4



1 tan2 t sec2 t

1 tan2 t 2 2

tan2 t 3

tan t 3

tan t 3 tan t 0 is given

Use Pythagorean identity involving sec t.

Solution continuedb. Given sec t 2 and tan t 0, find tan t.

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Slide 5.3- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley