494 CHAPTER 7 Analytic Trigonometry Finding All the Solutions of a Trigonometric Equation Solve the equation: θ = cos 1 2 Give a general formula for all the solutions. List eight of the solutions. Solution EXAMPLE 2 The period of the cosine function is π2 . In the interval π [ ) 0, 2 , there are two angles θ for which θ θ π = = cos 1 2 : 3 and θ π = 5 3 . See Figure 26(a). Because the cosine function has period π2 , all the solutions of θ = cos 1 2 may be given by the general formula θ π π θ π π = + = + k k k 3 2 or 5 3 2 any integer Eight of the solutions are π π π π π π π π − − 5 3 , 3 , 3 , 5 3 , 7 3 , 11 3 , 13 3 , 17 3 k 1 =− k 0 = k 1 = k 2 = Check: To verify the solutions, graph = y x cos and = y 1 2 and determine where the graphs intersect. (Be sure to graph in radian mode.) Figure 26(b) shows the graph using Desmos. One advantage to using Desmos is that the exact solutions are provided when the solution is a special angle (multiples of π 6 or π 4 ). If they are not special angles, then a decimal approximation will be provided. y x x2 1 y2 5 1 u 5 ( , y) (0, 1) (0, 21) (21, 0) (1, 0) p– 3 1 – 2 1 – 2 ( , 2y) u 5 5p ––– 3 (a) Figure 26 (b) Solving a Linear Trigonometric Equation Solve the equation: θ θ π + = ≤ < 2 sin 3 0, 0 2 Solution EXAMPLE 3 First solve the equation for θ sin . θ θ θ + = = − = − 2 sin 3 0 2 sin 3 sin 3 2 Subtract 3 from both sides. Divide both sides by 2. In the interval π [ ) 0, 2 , there are two angles θ for which θ θ π = − = sin 3 2 : 4 3 and θ π = 5 3 . The solution set is π π { } 4 3 , 5 3 . Now Work PROBLEM 37 In most of the work we do, we shall be interested only in finding solutions of trigonometric equations for θ π ≤ < 0 2 . CAUTION In Example 3, if a calculator is used to find − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − sin 3 2 , 1 the result given is ≈−1.047, which is not in the interval π [ ) 0, 2 . This occurs because, as we noted in Section 7.1, the range of = − y x sin 1 is π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 , so the calculator provides output in this range. This limitation of the calculator highlights the importance of knowing the common angles and the values of the six trigonometric functions of these angles. j Now Work PROBLEM 13
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