556 CHAPTER 5 Trigonometric Functions x y 0 u = 6758 u9 = 458 Figure 37 (b) Subtract 360° to find an angle between 0° and 360° coterminal with 675°. 675° - 360° = 315° As shown in Figure 37, the reference angle is 360° - 315° = 45°. An angle of 315° is in quadrant IV, so the tangent will be negative. tan 675° = tan 315° Coterminal angle = - tan 45° Reference angle; quadrant-based sign choice = -1 Evaluate. S Now Try Exercises 89 and 91. Determination of Angle Measures with Special Reference Angles The ideas discussed in this section can be used “in reverse” to find the measures of certain angles, given a trigonometric function value and an interval in which the angle must lie. We are most often interested in the interval 30°, 360°2. Degree mode A calculator can be used to find exact values such as cos 1-240°2 and tan 675°. Calculator Approximations of Trigonometric Function Values We have found exact function values for special angles and for angles having special reference angles. Calculators provide approximations for function values of angles that do not satisfy these conditions. EXAMPLE 7 Finding Angle Measures Find all values of u, if u is in the interval 30°, 360°2 and cos u = - 22 2 . SOLUTION The value of cos u is negative, so u may lie in either quadrant II or III. Because the absolute value of cos u is 22 2 , the reference angle u′ must be 45°. The two possible angles u are sketched in Figure 38. 180° - 45° = 135° Quadrant II angle u (from Figure 38(a)) 180° + 45° = 225° Quadrant III angle u (from Figure 38(b)) S Now Try Exercise 97. x y 0 u9 = 458 u in quadrant II u = 1358 (a) x y u in quadrant III 0 u = 2258 u9 = 458 (b) Figure 38 CAUTION When evaluating trigonometric functions of angles given in degrees, the calculator must be in degree mode. An easy way to check this is to enter sin 90. The displayed answer should be 1. Also, if the angle or the reference angle is not a special or quadrantal angle, then the value given by the calculator is an approximation. And even if the angle or reference angle is a special angle, the value given by the calculator will often be an approximation.
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