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18 3. EVALUATION of TRIGONOMETRIC FUNCTIONS in This Section, We Obtain Values of the Trigonometric Functions for Quadrantal

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18 3. EVALUATION of TRIGONOMETRIC FUNCTIONS in This Section, We Obtain Values of the Trigonometric Functions for Quadrantal 3. EVALUATION OF TRIGONOMETRIC FUNCTIONS In this section, we obtain values of the trigonometric functions for quadrantal angles, we introduce the idea of reference angles, and we discuss the use of a calculator to evaluate trigonometric functions of general angles. In Definition 2.1, the domain of each trigonometric function consists of all angles θ for which the denominator in the corresponding ratio is not zero. Because r > 0, it follows that sinθ = y/r and cosθ = x/r are defined for all angles θ . However, tanθ = y/x and secθ = r/x are not defined when the terminal side of θ lies anlong the y axis (so that x = 0). Likewise, cotθ = x/y and cscθ = r/y are not defined when the terminal side of θ lies along the x axis (so that y = 0). Therefore, when you deal with a trigonometric function of a quadrantal angle, you must check to be sure that the function is actually defined for that angle. Example 3.1 ---------------------------- ------------------------------------------------------------ Find the values (if they are defined) of the six trigonometric functions for the quadrantal angle θ = 90° (or θ = π 2 ). In order to use Definition 1, we begin by choosing any point ( 0 , y ) with y > 0, on the terminal side of the 90° angle (Figure 1). Because x = 0, it follows that tan 90° and sec 90° are undefined. Since y > 0, we have r = x2 + y 2 = 02 + y 2 = y 2 = y = y. y y x 0 Therefore, sin 90° = = = 1 cos 90° = = = 0 r y r y r y x 0 csc90° = = = 1 cot 90° = = = 0. y y y y _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ The values of the trigonometric functions for other quadrantal angles are found in a similar manner. The results appear in Table 3.1. Dashes in the table indicate that the function is undefined for that angle. Table 3.1 θ degrees θ radians sinθ cosθ tanθ cotθ secθ cscθ 0° 0 0 1 0 –– 1 –– π 90° 1 0 –– 0 –– 1 2 180° π 0 –1 0 –– –1 –– 3π 270° –1 0 –– 0 –– –1 2 360° 2π 0 1 0 –– 1 –– 18 It follows from Definition 2.1 that the values of each of the six trigonometric functions remain unchanged if the angle is replaced by a coterminal angle. If an angle exceeds one revolution or is negative, you can change it to a nonnegative coterminal angle that is less than one revolution by adding or subtracting an integer multiple of 360° (or 2π radians). For instance, sin 450° = sin( 450° − 360° ) = sin 90° = 1. sec 7π = sec () 7π − () 3 × 2π = secπ = –1. 1 cos ()− 660° = cos ()− 660° + (2 × 360°) = cos 60° = . 2 In Examples 3.2 and 3.3, replace each angle by a nonnegative coterminal angle that is less than on revolution and then find the values of the six trigonometric functions (if they are defined). Example 3.2 ---------------------------- ------------------------------------------------------------ θ = 1110° By dividing 1110 by 360, we find that the largest integer multiple of 360° that is less than 1110° is 3 × 360° = 1080° . Thus, 1110° – ( 3 × 360° ) = 1110° – 1080° = 30° . (Or we could have started with 1110° and repeatedly subtracted 360° until we obtained 30° .) It follows that 1 sin1110° = sin 30° = csc1110° = csc 30° = 2 2 3 2 3 cos1110° = cos 30° = sec1110° = sec 30° = 2 3 3 tan1110° = tan 30° = cot1110° = cot 30° = 3 3 ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Example 3.3 ---------------------------- ------------------------------------------------------------ 5π θ = – 2 5π We repeatedly add 2π to – until we obtain a nonnegative coterminal angle: 2 5π π – + 2π = – (still negative) 2 2 π 3π – + 2π = . 2 2 Therefore, by Table 3.1 for quadrantal angles, ⎛ 5π ⎞ ⎛ 3π ⎞ ⎛ 5π ⎞ ⎛ 3π ⎞ sin ⎜− ⎟ = sin ⎜ ⎟ = –1 cot ⎜− ⎟ = cot ⎜ ⎟ = 0 ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛ 5π ⎞ ⎛ 3π ⎞ ⎛ 5π ⎞ ⎛ 3π ⎞ cos ⎜− ⎟ = cos ⎜ ⎟ = 0 csc ⎜− ⎟ = csc ⎜ ⎟ = –1 ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛ 5π ⎞ ⎛ 5π ⎞ and both tan ⎜− ⎟ and sec ⎜− ⎟ are undefined. ⎝ 2 ⎠ ⎝ 2 ⎠ ______________________________________________________________________________________ 19 Table 3.1 θ degrees θ radians sinθ cosθ tanθ cotθ secθ cscθ π 1 3 3 2 3 30° 3 2 6 2 2 3 3 π 2 2 45° 1 1 2 2 4 2 2 π 3 1 3 2 3 60° 3 2 3 2 2 3 3 Figure 3.2 y y θ = 180° ‐ θ R θ = π ‐ θ R θ θR θ = θR O O x x (a) (b) y y θR = θ ‐ 180° θR = 360° ‐ θ θ θ π θ π θ R = ‐ R = 2 ‐ θ θ O x O x θR θ R (c) (d) 20 Example 3.4 ---------------------------- ------------------------------------------------------------ Find the reference angle θ R for each angle θ . 3π 5π (a) θ = 60° (b) θ = (c) θ = 210° (d) θ = . 4 3 (a) By Figure 3.2(a), θ R = θ = 60° . 3π π (b) By Figure 3.2(b), θ = π – θ = π – = . R 4 4 (c) By Figure 3.2(c), θ R = θ – 180° = 210° – 180° = 30° . 5π π (d) By Figure 3.2(c), θ = 2π – θ = 2π – = . R 3 3 ______________________________________________________________________________________ The value of any trigonometric function of any angle θ is the same as the value of the function for the reference angle, θ R , except possibly for a change of algebraic sign. Thus, sinθ = ± sinθ R , cosθ = ± cosθ R , and so forth. You can always determine the correct algebraic sign by considering the quadrant in which θ lies. Section 3 Problems---------------------- ------------------------------------------------------------ In problems 1 and 2, find the values (if they are defined) of the six trigonometric functions of the given quadrantal angles. (Do not use a calculator.) 1. (a) 0° (b) 180° (c) 270° (d) 360° . [When you have finished, compare your answers with the results in Table 3.1] 5π 7π 2. (a) 5π (b) 6π (c) –7π (d) (e) . 2 2 In Problems 3 to 14, replace each angle by a nonnegative coterminal angle that is less than one revolution and then find the exact values of the six trigonometric functions (if they are defined) for the angle. 3. 1440° 4. 810° 5. 900° 6. – 220° 7. 750° 8. 1845° 19π 9. – 675° 10. 2 25π 11. 5π 12. 6 13. 17π 14. – 31π 3 4 21 15. What happens when you try to evaluate tan 900° on a calculator? [Try it.] 16. Let θ be a quadrant III angle in standard position and let θ R be its reference angle. Show that the value of any trigonometric function of θ is the same as the value of θ R , except possibly for a change of algebraic sign. Repeat for θ in quadrant IV. In problems 17 to 36, find the reference angle θR for each angle θ , and then find the exact values of the six trigonometric functions of θ . 17. θ = 150° 18. θ = 120° 19. θ = 240° 20. θ = 225° 21. θ = 315° 22. θ = 675° 5π 23. θ = –150° 24. θ = – 6 13π 25. θ = – 60° 26. θ = – 6 π 53π 27. θ = – 28. θ = 4 6 2π 9π 29. θ = – 30. θ = 3 4 7π 50π 31. θ = 32. θ = – 4 3 11π 147π 33. θ = 34. θ = – 3 4 35. θ = – 420° 36. θ = – 5370° 37. Complete the following tables. (Do not use a calculator.) θ degrees θ radians sinθ cosθ tanθ 7π 210° 6 5π 225° 4 4π 240° 3 5π 300° 3 7π 315° 4 11π 330° 6 22 θ degrees θ radians cotθ secθ cscθ 7π 210° 6 5π 225° 4 4π 240° 3 5π 300° 3 7π 315° 4 11π 330° 6 38. A calculator is set in radian mode. π is entered and the sine (SIN) key is pressed. The display shows – 4.1 × 10-10 . But we know that sinπ = 0. Explain. In problems 59 to 62, use a calculator to verify that the equation is true for the indicated value of the angle θ . sinθ 59. tanθ = for θ = 35° . cosθ 5π 60. (cosθ )(tanθ ) = sinθ for θ = 7 5π 61. cos2 θ + sin 2 θ = 1 for θ = 3 62. 1+ tan 2 θ = sec2 θ for θ = 17.75° 0 1 2 3 4 63. Verify that for θ = 0° , 30° , 45° , 60° , 90° , we have sinθ = , , , , and 2 2 2 2 2 respectively. [Although there is no theoretical significance to this pattern, people often use it as a memory aid to help recall these values of sinθ .] 23.
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