5 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley

Trigonometric Functions 5 5.1 Angles 5.2 Trigonometric Functions 5.3 Evaluating Trigonometric Functions 5.4 Solving Right Triangles Copyright © 2009 Pearson Addison-Wesley

Evaluating Trigonometric Functions 5.3 Evaluating Trigonometric Functions Right-Triangle-Based Definitions of the Trigonometric Functions ▪ Cofunctions ▪ Trigonometric Function Values of Special Angles ▪ Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Function Values Using a Calculator ▪ Finding Angle Measures with Special Angles Copyright © 2009 Pearson Addison-Wesley 1.1-3

Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position, Copyright © 2009 Pearson Addison-Wesley 1.1-4

Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position, Copyright © 2009 Pearson Addison-Wesley 1.1-5

Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position, Copyright © 2009 Pearson Addison-Wesley 1.1-6

Find the sine, cosine, and tangent values for angles A and B. Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE Find the sine, cosine, and tangent values for angles A and B. Copyright © 2009 Pearson Addison-Wesley 1.1-7

Find the sine, cosine, and tangent values for angles A and B. Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE (cont.) Find the sine, cosine, and tangent values for angles A and B. Copyright © 2009 Pearson Addison-Wesley 1.1-8

Cofunction Identities For any acute angle A in standard position, sin A = cos(90  A) csc A = sec(90  A) tan A = cot(90  A) cos A = sin(90  A) sec A = csc(90  A) cot A = tan(90  A) Copyright © 2009 Pearson Addison-Wesley 1.1-9

Write each function in terms of its cofunction. Example 2 WRITING FUNCTIONS IN TERMS OF COFUNCTIONS Write each function in terms of its cofunction. (a) cos 52° = sin (90° – 52°) = sin 38° (b) tan 71° = cot (90° – 71°) = cot 19° (c) sec 24° = csc (90° – 24°) = csc 66° Copyright © 2009 Pearson Addison-Wesley 1.1-10

30°- 60°- 90° Triangles Bisect one angle of an equilateral to create two 30°-60°-90° triangles. Copyright © 2009 Pearson Addison-Wesley

30°- 60°- 90° Triangles Use the Pythagorean theorem to solve for x. What would x have been if the shorter side had been 5? Copyright © 2009 Pearson Addison-Wesley

Find the six trigonometric function values for a 60° angle. Example FINDING TRIGONOMETRIC FUNCTION VALUES FOR 60° Find the six trigonometric function values for a 60° angle. Copyright © 2009 Pearson Addison-Wesley 1.1-13

Find the six trigonometric function values for a 60° angle. Example FINDING TRIGONOMETRIC FUNCTION VALUES FOR 60° (continued) Find the six trigonometric function values for a 60° angle. Copyright © 2009 Pearson Addison-Wesley 1.1-14

45°- 45° Right Triangles Use the Pythagorean theorem to solve for r. Copyright © 2009 Pearson Addison-Wesley

45°- 45° Right Triangles Do the trig functions depend on the length of the side? Copyright © 2009 Pearson Addison-Wesley

45°- 45° Right Triangles Copyright © 2009 Pearson Addison-Wesley

Function Values of Special Angles  sin  cos  tan  cot  sec  csc  30 45 60 Copyright © 2009 Pearson Addison-Wesley

Reference Angles A reference angle for an angle θ is the positive acute angle made by the terminal side of angle θ and the x-axis. Copyright © 2009 Pearson Addison-Wesley

The reference angle is always found with reference to the x-axis. Caution A common error is to find the reference angle by using the terminal side of θ and the y-axis. The reference angle is always found with reference to the x-axis. Copyright © 2009 Pearson Addison-Wesley 1.1-20

For θ = 218°, the reference angle θ′ = 38°. Example 3(a) FINDING REFERENCE ANGLES Find the reference angle for an angle of 218°. The positive acute angle made by the terminal side of the angle and the x-axis is 218° – 180° = 38°. For θ = 218°, the reference angle θ′ = 38°. Copyright © 2009 Pearson Addison-Wesley 1.1-21

Find the reference angle for an angle of 1387°. Example 3(b) FINDING REFERENCE ANGLES Find the reference angle for an angle of 1387°. First find a coterminal angle between 0° and 360°. Divide 1387 by 360 to get a quotient of about 3.9. Begin by subtracting 360° three times. 1387° – 3(360°) = 307°. The reference angle for 307° (and thus for 1387°) is 360° – 307° = 53°. Copyright © 2009 Pearson Addison-Wesley 1.1-22

Copyright © 2009 Pearson Addison-Wesley 1.1-23

Find the values of the six trigonometric functions for 210°. Example 4 FINDING TRIGONOMETRIC FUNCTION VALUES OF A QUADRANT III ANGLE Find the values of the six trigonometric functions for 210°. The reference angle for a 210° angle is 210° – 180° = 30°. Choose point P on the terminal side of the angle so the distance from the origin to P is 2. Copyright © 2009 Pearson Addison-Wesley 1.1-24

Example 4 FINDING TRIGONOMETRIC FUNCTION VALUES OF A QUADRANT III ANGLE (continued) Copyright © 2009 Pearson Addison-Wesley 1.1-25

Find the exact value of cos (–240°). Example 5(a) FINDING TRIGONOMETRIC FUNCTION VALUES USING REFERENCE ANGLES Find the exact value of cos (–240°). Since an angle of –240° is coterminal with an angle of –240° + 360° = 120°, the reference angle is 180° – 120° = 60°. Copyright © 2009 Pearson Addison-Wesley 1.1-26

Find the exact value of tan 675°. Example 5(b) FINDING TRIGONOMETRIC FUNCTION VALUES USING REFERENCE ANGLES Find the exact value of tan 675°. Subtract 360° to find a coterminal angle between 0° and 360°: 675° – 360° = 315°. Copyright © 2009 Pearson Addison-Wesley 1.1-27

Caution When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode. Copyright © 2009 Pearson Addison-Wesley 1.1-28

Approximate the value of each expression. Example 6 FINDING FUNCTION VALUES WITH A CALCULATOR Approximate the value of each expression. (a) sin 53° (b) sec 97.977° Calculators do not have a secant key, so first find cos 97.977° and then take the reciprocal. sec 97.977° ≈ –7.20587921 Copyright © 2009 Pearson Addison-Wesley 1.1-29

Approximate the value of each expression. Example 6 FINDING FUNCTION VALUES WITH A CALCULATOR (continued) Approximate the value of each expression. (c) Use the reciprocal identity Copyright © 2009 Pearson Addison-Wesley 1.1-30

cos θ = .5 We know that cos ____ = .5 Example 7 USING INVERSE TRIGONOMETRIC FUNCTIONS TO FIND ANGLES Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition. cos θ = .5 We know that cos ____ = .5 To find this on the calculator we use θ = b) Find θ if cos θ = .9211854056 c) Find θ if sec θ = 1.2228 Copyright © 2009 Pearson Addison-Wesley 1.1-31

Find all values of θ, if θ is in the interval [0°, 360°) and Example 8 FINDING ANGLE MEASURES GIVEN AN INTERVAL AND A FUNCTION VALUE Find all values of θ, if θ is in the interval [0°, 360°) and Since cos θ is negative, θ must lie in quadrant II or III. The absolute value of cos θ is so the reference angle is 45°. The angle in quadrant II is 180° – 45° = 135°. The angle in quadrant III is 180° + 45° = 225°. Copyright © 2009 Pearson Addison-Wesley 1.1-32

Example 9 Problem 141, page 529 When highway curves are designed, the outside of the curve is often slightly elevated or inclined above the inside of the curve. This inclination is called superelevation . For safety reasons, it is important that both the curve’s radius and superelevation are correct for a given speed limit. If an automobile is traveling at velocity V (in feet per second), the safe radius R for a curve with superelevation θ is modeled by the formula where f and g are constants

Example 9 A roadway is being designed for automobiles traveling at 45mph. If θ = 3°, g = 32.2, and f = 0.14, calculate R to the nearest foot. (Hint: You will need to know that 1 mile = 5280 feet) b) A highway curve has radius R = 1150 ft and a superelevation of θ = 2.1°. What should the speed limit (in miles per hour) be for this curve? Use the same values for f and g from part (a).