406 CHAPTER 6 Trigonometric Functions Now Work PROBLEM 65 Using a Calculator to Approximate the Value of a Trigonometric Function Use a calculator to find the approximate value of: (a) cos48° (b) csc21° (c) tan 12 π Express your answer rounded to two decimal places. EXAMPLE 11 Solution (a) First, set the MODE to receive degrees. Figure 33(a) shows the solution using a TI-84 Plus CE graphing calculator. Rounded to two decimal places, cos48 0.67 ° = (b) Most calculators do not have a csc key. The manufacturers assume that the user knows some trigonometry. To find the value of csc 21 ,° use the fact that csc21 1 sin21 . ° = ° Figure 33(b) shows the solution using a TI-84 Plus CE graphing calculator. Rounded to two decimal places, csc21 2.79 ° = (c) Set the MODE to receive radians. Figure 33(c) shows the solution using a TI-84 Plus CE graphing calculator. Rounded to two decimal places, tan 12 0.27 π = Figure 34 (a) (b) Figure 34(a) shows the results of Example 11(a) and 11(b) using Desmos. Notice that Desmos does have a csc function. Use the wrench icon in Desmos to switch between degree and radian mode. Figure 34(b) shows the results of Example 11(c) using Desmos. Figure 33 (a) cos48° (b) csc21° (c) π tan 12 Degree mode Radian mode 7 Use a Circle of Radius r to Evaluate the Trigonometric Functions Until now, finding the exact value of a trigonometric function of an angle θ required that we locate the corresponding point P x y , ( ) = on the unit circle. In fact, though, any circle whose center is at the origin can be used. Let θ be any nonquadrantal angle placed in standard position. Let P x y , ( ) = be the point on the circle x y r 2 2 2 + = that corresponds to ,θ and let ( ) = P x y * *, * be the point on the unit circle that corresponds to .θ See Figure 35, where θ is shown in quadrant II. Figure 35 y u P* 5 (x*, y*) P 5 (x, y) x O r A x A* x* y* 1 x2 1 y2 5 1 x2 1 y2 5 r 2 y
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