1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 4 Trigonometric Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 The Unit Circle; Trigonometric Functions of an Angle Define the trigonometric functions using the unit circle. Find exact trigonometric function values using a point on the unit circle. Find trigonometric function values of quadrantal angles. Find trigonometric function values of any angle. Approximate trigonometric function values using a calculator. SECTION

3 © 2010 Pearson Education, Inc. All rights reserved In a unit circle, r = 1; so the length, s, of the intercepted arc is s = 1 ∙ θ or s = θ. That is, the radian measure and the arc length are identical. THE UNIT CIRCLE

4 © 2010 Pearson Education, Inc. All rights reserved

5 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

6 © 2010 Pearson Education, Inc. All rights reserved A point P on the unit circle associated with a real number t has coordinates (cos t, sin t) because x = cos t and y = sin t. POINTS ON THE UNIT CIRCLE

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Evaluating Trigonometric Functions Find the values (if any) of the six trigonometric functions of each value of t.

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Evaluating Trigonometric Functions a. t = 0 corresponds to the point (x, y) = (1, 0). Solution

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Evaluating Trigonometric Functions b. Solution continued

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Evaluating Trigonometric Functions c. t = π corresponds to the point (x, y) = (−1, 0). Solution continued

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Evaluating Trigonometric Functions d. Solution continued

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Evaluating Trigonometric Functions e. t = −3π corresponds to the same point, (−1, 0), as t = π. Solution continued

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14 © 2010 Pearson Education, Inc. All rights reserved Given an angle θ in standard position, let P(x, y) be the point where the terminal ray of θ intersects the unit circle. TRIGONOMETRIC FUNCTIONS OF AN ANGLE

15 © 2010 Pearson Education, Inc. All rights reserved If θ is an angle with radian measure t, then TRIGONOMETRIC FUNCTIONS OF AN ANGLE If θ is given in degrees, convert θ to radians before using these equations.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding the Trigonometric Function Values of a Quadrantal Angle Find the trigonometric function values of 90º. Solution, so

17 © 2010 Pearson Education, Inc. All rights reserved TRIGONOMETRIC FUNDTIONS OF QUADRANTAL ANGLES

18 © 2010 Pearson Education, Inc. All rights reserved TRIGONOMETRIC FUNDTIONS OF QUADRANTAL ANGLES

19 © 2010 Pearson Education, Inc. All rights reserved QUADRANTAL ANGLES

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21 © 2010 Pearson Education, Inc. All rights reserved There’s every reason to draw a circle.

22 © 2010 Pearson Education, Inc. All rights reserved TRIGONOMETRIC VALUES OF AN ANGLE θ

23 © 2010 Pearson Education, Inc. All rights reserved VALUES OF TRIGONOMETRIC VALUES OF AN ANGLE θ Let P(x, y) be any point on the terminal ray of an angle in standard position (other than the origin) and let r = Then r > 0, and

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 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

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Finding Trigonometric Function Values Solution continued

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27 © 2010 Pearson Education, Inc. All rights reserved As a note on exact values, it is always better to use these throughout a general evaluation and only round your result. Calculators are not always correct. You should certainly be able to determine the lengths of a right triangle with angles of 45 degrees and 30 and 60 …

28 © 2010 Pearson Education, Inc. All rights reserved TRIGONOMETRIC FUNCTION VALUES FOR AND

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding Exact Trigonometric Function Values of Find the exact trigonometric function values of Solution The point (x, y) = is on the terminal side of

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding Exact Trigonometric Function Values of Solution continued

31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding Exact Trigonometric Function Values of Solution continued

32 © 2010 Pearson Education, Inc. All rights reserved MORE TRIGONOMETRIC FUNCTION VALUES

33 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding Chord Length on the Unit Circle Find the length of the chord of the unit circle intercepted by an angle of  radians. Solution y = = half the length of the chord. So, = 2y = length of chord

34 © 2010 Pearson Education, Inc. All rights reserved TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES These equations hold for any integer n.

35 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Trigonometric Function Values of Coterminal Angles Find the exact values for a. sin 2580ºb. Solution a.2580° = 60° ° = 60° + 7(360°); so sin 2580º = sin 60º = b. so

36 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Approximating Trigonometric Function Values Using a Calculator Use a calculator to find the approximate value of each expression. Round your answers to two decimal places. a. sin 71ºb. tanc. sec 1.3 Solution a.Set the MODE to degrees. sin 71º ≈ ≈ 0.95

37 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Approximating Trigonometric Function Values Using a Calculator Solution continued c. Set the MODE to radians. sec 1.3 = ≈ ≈ 3.74 b. Set the MODE to radians. tan ≈ − ≈ −1.25