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Trigonometric equations

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Trigonometric equations

  1. 1. Trigonometric EquationsFind the general solutions (express in radian) of following equations 2n 1. 2 cos(3x ) 1 0 Ans. n 6 3 12 18 3x 4n 2 2. 2 sin 1 0 Ans. ( 1) n 1 n 4 3 9 x 3. tan( ) 3 Ans. 4n n 4 3 4. sin x 1 cos x Ans. n ( 1) n n 4 4 5. 3 sin x cos x 3 Ans. n ( 1) n n 3 6 6. ( 3 1) cos x ( 3 1) sin x 2 Ans. 2n n 4 12 x 2 7. sin x sin Ans. 2n , 4n 2 3 n 2n 8. 3sin 7 x 2sin 4x 3sin x 0 Ans. , 0.4103 4 3 9. tan( ) tan( ) 4 Ans. n 4 4 6 (4n 1) (4n 1) 10. cos3 sin Ans. , 8 4 Find the general solutions of the following equations and gives where . n (2n 1) 2 3 3 4 11. sin 7 x sin3x 0 Ans. , ; 0, , , , , , , 5 4 5 4 5 5 4 5 2n 2 12. sin x sin 2x sin3x Ans. , n ; 0, , 3 3 n ( 1)n 3 4 ; 5 29 13. sin(2 x ) cos( x) Ans. n 1 , , 4 6 2 ( 1) 12 36 36 n n 5 7 3 11 14. sin 4x cos 2 x 0 Ans. , ( 1)n 1 ; , , , , , 2 4 2 12 12 12 4 4 12 4 12 15. 3sin 2x 2tan x Ans. n , n 0.9553 ; 0,0.9553, 2.1866,
  2. 2. Find the general solutions of the following equations. 316. 2sin3x cos 2x 1 Ans. n , n ( 1)n 0.8481, n ( 1) n 217. 5cos 2x 8sin x 3 Ans. n ( 1) n 0.2014 , 2n 2 2n 2n n18. sin5x sin3x sin8x Ans. , , 5 10 3 6 419. 2 cos 2 3 sin 1 0 Ans. n ( 1)n 1 3 3 n20. cos3 x sin x sin 3 x cos x Ans. ( 1) n 8 4 1221. 4cos 3sin 3 Ans. 2n 0.9273 0.643522. cos 2 x 4sin 2 x cos x 2 0 Ans. (2n 1) , 2n 3 n (2n 1)23. sin 2 sin 2 2 sin 2 3 Ans. , 2 624. a) Prove that cos3 4cos3 3cos . b) Solve the equation cos 3 cos ,where 0 6 11 13 Ans. , , 18 18 18 c) By the results of questions a) and b) ,deduce the roots of the equation 3 4 x3 3x .Correct the answer to 3 decimal places. 2 Ans. 0.985, 0.342, 0.643

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