748 CHAPTER 7 Trigonometric Identities and Equations 7.6 Exercises CONCEPT PREVIEWUse the unit circle shown here to solve each simple trignometric equation. If the variable is x, then solve over 30, 2p2. If the variable is u, then solve over 30°, 360°2. 1. cos x = 1 2 2. cos x = 23 2 3. sinx = - 1 2 4. sinu = 0 5. sinu = -1 6. cos u = - 22 2 (0, 1) (1, 0) (0, –1) (–1, 0) 0 08 1808 608 908 1508 2108 3008 3608 3158 3308 2P P 1358 1208 2258 2408 270° 458 308 1 2 ( , ) √3 2 1 2( , – ) √3 2 1 2 (– , – ) √3 2 ( , ) √2 2 √2 2 1 2 ( , – ) √3 2 1 2 ( , ) √3 2 1 2 (– , ) √3 2 (– , ) √2 2 √2 2 ( , – ) √2 2 √2 2 1 2 (– , – ) √3 2 (– , – ) √2 2 √2 2 1 2 (– , ) √3 2 3P 2 5P 3 7P 4 4P 3 5P 4 7P 6 5P 6 3P 4 2P 3 11P 6 P 3 P 2 P 4 P 6 x y 0 CONCEPT PREVIEWSolve for exact solutions over the interval 30, 2p2. 7. cos 2x = 1 2 8. cos 2x = 23 2 9. sin2x = - 1 2 CONCEPT PREVIEWSolve for exact solutions over the interval 30°, 360°2. 10. sin u 2 = 0 11. sin u 2 = -1 12. cos u 2 = - 12 2 13. Concept Check Suppose that in solving an equation over the interval 30°, 360°2, we reach the step sinu = - 1 2. Why is -30° not a correct answer? 14. Concept Check Lindsay solved the equation sinx = 1 - cos x by squaring each side to obtain sin2 x = 1 - 2 cos x + cos2 x. Several steps later, using correct algebra, she concluded that the solution set for solutions over the interval 30, 2p2 is E0, p 2 , 3p 2 F. Explain why this is not correct. Solve each equation for exact solutions over the interval 30, 2p2. See Examples 1–3. 15. 2 cot x + 1 = -1 16. sinx + 2 = 3 17. 2 sinx + 3 = 4 18. 2 sec x + 1 = sec x + 3 19. tan2 x + 3 = 0 20. sec2 x + 2 = -1 21. 1cot x - 12A 23 cot x + 1B = 0 22. 1csc x + 22Acsc x - 22B = 0 23. cos2 x + 2 cos x + 1 = 0 24. 2 cos2 x - 23 cos x = 0 25. -2 sin2 x = 3 sinx + 1 26. 2 cos2 x - cos x = 1 Solve each equation over the interval 30°, 360°2. Write solutions as exact values or to the nearest tenth, as appropriate. See Examples 2–5. 27. Acot u - 23B A2 sinu + 23B = 0 28. 1tanu - 121cos u - 12 = 0 29. 2 sinu - 1 = csc u 30. tanu + 1 = 23 + 23 cot u 31. tanu - cot u = 0 32. cos2 u = sin2 u + 1
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