552 CHAPTER 5 Trigonometric Functions Function Values of Special Angles U sinU cos U tanU cot U sec U csc U 30° 1 2 !3 2 !3 3 ! 3 2!3 3 2 45° ! 2 2 !2 2 1 1 !2 !2 60° ! 3 2 1 2 ! 3 ! 3 3 2 2!3 3 Reference Angles Associated with every nonquadrantal angle in standard position is an acute angle called its reference angle. A reference angle for an angle u, written u′, is the acute angle made by the terminal side of angle u and the x-axis. We find the values of the trigonometric functions for 45° by starting with a 45°945° right triangle, as shown in Figure 32. This triangle is isosceles. For simplicity, we choose the lengths of the equal sides to be 1 unit. (As before, the results are independent of the length of the equal sides.) If r represents the length of the hypotenuse, then we can find its value using the Pythagorean theorem. 12 + 12 = r2 Pythagorean theorem 2 = r2 Apply the exponents and add. 2 2 = r Choose the positive root. Now we use the measures indicated on the 45°945° right triangle in Figure 32. sin 45° = 12 2 = 22 2 cos 45° = 12 2 = 22 2 tan 45° = 1 1 = 1 csc 45° = 22 1 = 22 sec 45° = 22 1 = 22 cot 45° = 1 1 = 1 Function values for 30°, 45°, and 60° are summarized in the table that follows. Figure 32 458 458 1 r = Á2 1 45°–45° right triangle Figure 33 on the next page shows several angles u (each less than one complete counterclockwise revolution) in quadrants II, III, and IV, respectively, with the reference angle u′ also shown. In quadrant I, angles u and u′ are the same. If an angle u is negative or has measure greater than 360°, its reference angle is found by first finding its coterminal angle that is between 0° and 360°, and then using the diagrams in Figure 33. NOTE To reproduce this table quickly, it is important to learn the values of sin 30°, sin 45°, and sin 60°. Then the rest of the table can be completed using the reciprocal, cofunction, and quotient identities. NOTE Reference angles are always positive and are between 0° and 90°. NOTE The results in Example 3 can also be found using the fact that cofunction values of the complementary angles 60° and 30° are equal.
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