537 5.2 Trigonometric Functions Recall that when the equation of a line is written in the form y = mx + b, Slope-intercept form the coefficient m of x gives the slope of the line. In Example 3, the equation x + 2y = 0 can be solved for y to obtain y = - 1 2 x. The slope of this line is - 1 2 . Notice also that tan u = - 1 2 . In general, it is true that m=tan U. Quadrantal Angles If the terminal side of an angle in standard position lies along the y-axis, any point on this terminal side has x-coordinate 0. Similarly, an angle with terminal side on the x-axis has y-coordinate 0 for any point on the terminal side. Because the values of x and y appear in the denominators of some trigonometric functions, and because a fraction is undefined if its denominator is 0, some trigonometric function values of quadrantal angles (i.e., those with terminal side on an axis) are undefined. When determining trigonometric function values of quadrantal angles, Figure 19 can help find the ratios. Because any point on the terminal side can be used, it is convenient to choose the point one unit from the origin, with r = 1. (Later we will extend this idea to the unit circle.) To find the function values of a quadrantal angle, determine the position of the terminal side, choose the one of these four points that lies on this terminal side, and then use the definitions involving x, y, and r. EXAMPLE 4 Finding Function Values of Quadrantal Angles Find the values of the six trigonometric functions for each angle. (a) an angle of 90° (b) an angle u in standard position with terminal side passing through 1-3, 02 SOLUTION (a) Figure 20 shows that the terminal side passes through 10, 12. So x = 0, y = 1, and r = 1. Thus, we have the following. sin 90° = 1 1 = 1 cos 90° = 0 1 = 0 tan 90° = 1 0 Undefined csc 90° = 1 1 = 1 sec 90° = 1 0 Undefined cot 90° = 0 1 = 0 x y 0 (–1, 0) (1, 0) (0, 1) (0, –1) x = 0 y = 1 r = 1 x = 1 y = 0 r = 1 x = –1 y = 0 r = 1 x = 0 y = –1 r = 1 Figure 19 A calculator in degree mode returns the correct values for sin 90° and cos 90°. The screen shows an ERROR message for tan 90°, because 90° is not in the domain of the tangent function. x y 0 (0, 1) 90° Figure 20 NOTE The trigonometric function values we found in Examples 1–3 are exact. If we were to use a calculator to approximate these values, the decimal results would not be acceptable if exact values were required.
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