496 CHAPTER 7 Analytic Trigonometry 2 Solve Trigonometric Equations Using a Calculator The next example illustrates how to solve trigonometric equations using a calculator. Remember that the function keys on a calculator give only values consistent with the definition of the function. Solving a Trigonometric Equation Using a Calculator Use a calculator to solve the equation θ θ π = − ≤ < tan 2, 0 2 . Express any solutions in radians, rounded to two decimal places. Solution EXAMPLE 6 To solve θ = − tan 2 on a calculator, first set the mode to radians.Then use the − tan 1 key to obtain θ ( ) = − ≈ − − tan 2 1.1071487 1 Rounded to two decimal places, θ ( ) = − = − − tan 2 1.11 1 radian. Because of the definition of = − y x tan , 1 the angle θ that is obtained is the angle π θ π − < < 2 2 for which θ = − tan 2. But we want solutions for which θ π ≤ < 0 2 . Since the period of the tangent function is π, the angles π −1.11 and π − 2 1.11 are solutions that lie in the interval π [ ) 0, 2 . Note that the angle π − 3 1.11 lies outside the interval and so is not a solution. The solutions of the equation θ θ π = − ≤ < tan 2, 0 2 , are θ π θ π = − ≈ = − ≈ 1.11 2.03 radians and 2 1.11 5.17 radians The solution set is { } 2.03, 5.17 . Figure 28 illustrates another way to obtain the solutions. Start with the angle θ = −1.11. Then π −1.11 is the angle in quadrant II, where θ = − tan 2, and π − 2 1.11 is the angle in quadrant IV where θ = − tan 2. Figure 28 θ =− tan 2 (21, 2) (1, 22) x y 2 1 –1 –2 –1 1 2 –2 u = p21.11 u = 2p21.11 u = 21.11 θ π π π θ π π − = + = + k k k 2 4 any integer 3 4 In the interval π θ π [ ) = 0, 2 , 3 4 and θ π π π = + = 3 4 7 4 are the only solutions. The solution set is π π { } 3 4 , 7 4 . Now Work PROBLEM 27 CAUTION Example 6 illustrates that caution must be exercised when solving trigonometric equations on a calculator. Remember that the calculator supplies an angle only within the restrictions of the definition of the inverse trigonometric function. To find the remaining solutions, you must identify other quadrants, if any, in which a solution may be located. j Now Work PROBLEM 49 3 Solve Trigonometric Equations Quadratic in Form Many trigonometric equations can be solved using techniques that we already know, such as using the quadratic formula (if the equation is a second-degree polynomial) or factoring.
RkJQdWJsaXNoZXIy NjM5ODQ=