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Solve Trigonometric Equations The designer of a new model of skis must model their performance as a skier slides down a hill. The model involves trigonometric functions. Once the design is modelled, equations must be solved for various snow conditions and other factors to determine the expected performance of the skis. In this section, you will learn how to solve trigonometric equations with and without technology. Investigate How can you use trigonometric identities and special angles to solve a trigonometric equation? 1. Consider the trigonometric equation 2 sin x 1 0. Solve the equation for sin x . 2. Refer to your Special Angles and Trigonometric Ratios table. a) Determine an angle x in the interval [0, 2 π ] whose sine matches the sine in step 1. Use radian measure. b) Determine another angle x in the interval [0, 2 π ] that matches this sine. 3. Use the left side/right side method to verify that your solutions are valid. 4. Reflect Are these the only angles in the interval [0, 2 π ] that work? Justify your answer. 5. Graph the left side of the equation. • Ensure that the graphing calculator is in Radian mode. • Enter 2 sin x 1 using x . • Press y and select 7:ZTrig . 6. a) Use the Zero operation to determine the fi rst zero to the right of the origin. • Press O [CALC] to display the CALCULATE menu. • Select 2:zero . • Move the cursor to locations for the left bound, right bound, and guess, pressing e after each. b) How does this value compare to your solutions in step 2? Tools • Special Angles and Trigonometric Ratios table from Chapter 4 • graphing calculator 5.4 282 MHR • Advanced Functions • Chapter 5

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7. Use the Zero operation to determine any other solutions in the interval [0, 2 π ]. Compare them to your solutions in step 2. 8. Reflect Suppose that the right side of the equation in step 1 were not 0, but 2. Could you still find the solutions by graphing the left side? If so, explain how, and demonstrate using a graphing calculator. If not, how could you modify the procedure to yield the correct solutions? Demonstrate your modifications using the graphing calculator. Example 1 Solve a Quadratic Trigonometric Equation Determine the exact solutions for the trigonometric equation 2 sin 2 x 1 0 in the interval x [0, 2 π ]. Solution Method 1: Use Pencil and Paper 2 sin² x 1 0 2 sin² x 1 sin² x 1 _ 2 sin x 1 _ 2 Case 1: sin x 1 _ 2 The angles in the interval [0, 2 π ] that satisfy sin x 1 _ 2 are x π _ 4 and x 3 π _ 4 . Case 2: sin x 1 _ 2 The angles in the interval [0, 2 π ] that satisfy sin x 1 _ 2 are x 5 π _ 4 and x 7 π _ 4 . The solutions in the interval [0, 2 π ] are x π _ 4 , x 3 π _ 4 , x 5 π _ 4 , and x 7 π _ 4 . C O N N E C T I O N S In Chapter 4, Section 4.2, you learned how special triangles, the unit circle, and the CAST rule can be used to determine exact values for the trigonometric ratios of special angles and their multiples.
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