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Trig Functions of Special Angles Objectives To find exact values for the six trigonometric functions of special angles To find decimal approximations for the values of the six trigonometric functions of any angle Quadrantal Angles The first group of special angles are quadrantal angles. Quadrantal angles are angles whose terminal side lies on one of the axes, making it easy to find the trig functions. (0, 1) cos 90 = 0 sin 90 = 1 cos 180 = -1 sin 180 = 0 cos 270 = 0 sin 270 = -1 sin 360 = 0 cos 360 = 1 (1, 0) (-1, 0) (0, -1) Other Special Angles The other special angles can be found using the trig version of the Pythagorean Theorem. We will look at these values in the form of a table. q (in radians) q (in degrees) cos q sin q 0 p/6 p/4 p/3 p/2 2p/3 0 30° 45° 60° 1 √3/2 √2/2 ½ 0 ½ √2/2 √3/2 3p/4 5p/6 p 90° 120 135 150 180 0 -½ 1 √3/2 √2/2 -√2/2 -√3/2 ½ 0 0 You can also use what we know about reference angles and their signs in certain quadrants to find the values. Remember, the reference angle is the acute angle formed by the terminal side and the axis. Find each of the following values. cos 5p/6 terminal side is in quadrant II The reference angle for 5p/6 in quadrant II is p/6. cos is x/r. x in quadrant II is negative. Cos p/3 is √3/2, but since x is negative in quadrant II, then cos 5p/6 = -√3/2. p 6 5p 6 Find tan(-11p/4) -11p/4 is in quadrant 3 The reference angle for -11p/4 is p/4. tan is y/x. In quadrant 3 y and x are both negative Find p/4 in your chart. Since we don’t know x and y, we can look at sin and cos to find x and y. sin is y/r and cos is x/r sin p/4 = √2/2 and cos p/4 = √2/2, so x = √2 and y = √2/2 p 4 -11p 4 reference angle So tan p/4 = √2/√2 or 1. Find csc 29p/3 29p/3 is coterminal with 5p/3 in quadrant 4 csc is the reciprocal of sin Find sin p/3 in your chart sin p/3 = √3/2 p 3 Since csc is the reciprocal of sin csc p/3 = 2/√3, so csc 29p/3 = 2/√3 Reference angle When you rationalize the denominator you get csc 29p/3 = 2√3/3. Assignment Page 267 – 268 – # 15 – 35, 42, 44