555 5.3 Trigonometric Function Values and Angle Measures Notice in Example 5 that the trigonometric function values of 210° correspond in absolute value to those of its reference angle 30°. The signs are different for the sine, cosine, secant, and cosecant functions because 210° is a quadrant III angle. These results suggest a shortcut for finding the trigonometric function values of a non-acute angle, using the reference angle. In Example 5, the reference angle for 210° is 30°. Using the trigonometric function values of 30°, and choosing the correct signs for a quadrant III angle, we obtain the same results. We determine the values of the trigonometric functions for any nonquadrantal angle u as follows. Keep in mind that all function values are positive when the terminal side is in Quadrant I, the sine and cosecant are positive in Quadrant II, the tangent and cotangent are positive in Quadrant III, and the cosine and secant are positive in Quadrant IV. In other cases, the function values are negative. EXAMPLE 6 FindingTrigonometric Function Values Using Reference Angles Find the exact value of each expression. (a) cos1-240°2 (b) tan 675° SOLUTION (a) Because an angle of -240° is coterminal with an angle of -240° + 360° = 120°, the reference angle is 180° - 120° = 60°, as shown in Figure 36. The cosine is negative in quadrant II. cos1-240°2 = cos 120° Coterminal angle = -cos 60° Reference angle = - 1 2 Evaluate. x y 0 u9 = 608 u = –2408 Figure 36 Finding Trigonometric Function Values for Any Nonquadrantal Angle U Step 1 If u 7360°, or if u 60°, then find a coterminal angle by adding or subtracting 360° as many times as needed to obtain an angle greater than 0° but less than 360°. Step 2 Find the reference angle u′. Step 3 Find the trigonometric function values for reference angle u′. Step 4 Determine the correct signs for the values found in Step 3. (Use the table of signs given earlier in the text or the paragraph above, if necessary.) This gives the values of the trigonometric functions for angle u. NOTE To avoid sign errors when finding the trigonometric function values of an angle, sketch it in standard position. Include a reference triangle complete with appropriate values for x, y, and r as done in Figure 35.
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