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INSTRUCTOR: Dr. Bathi Kasturiarachi NAME: Math 11022 Spring 2008 Trigonometry REVIEW for Exam 2 { Sections (6.5, 7.1, 7.2, 7.3) 1. Find the exact value of arcsin(1) = 2. 2 Find the exact value of tan = 3. 1 tan 5 4 4 Find the exact value of tan sin 1 1 2 1 1 , sin y = =) y = 2 2 p 6 1 3 ) tan sin 1 = tan = 2 6 3 Let y = sin 4. 1 Find the exact value of tan cos 1 2 5 2 2 , cos y = =) draw a triangle 5 5 p 21 1 2 = tan y = ) tan cos 5 2 Let y = cos 5. 1 Find an algebraic expression in terms of x for the following. sin (arccos x) x =) draw a triangle 1 p ) sin (arccos x) = sin y = 1 x2 Let y = cos 6. 1 x , cos y = Find an algebraic expression in terms of x for the following. ! p 2 x 5 sec cos 1 2x p p x2 5 x2 5 1 Let y = cos , cos y = =) draw a triangle 2x 2x ! p x2 5 2x sec cos 1 = sec y = p 2x x2 5 1 7. Prove the following trigonometric identity. 1 1 1 + cos x 1 + cos x 2 csc2 x LHS = = 8. 1 + cos x + 1 cos x (1 cos x)(1 + cos x) 2 2 = 2 csc2 x = RHS = 1 cos2 x sin2 x Prove the following trigonometric identity. sin3 + cos3 (sin + cos )(1 sin cos ) 2 LHS = (sin + cos )[sin = (sin + cos )(1 9. sin cos + cos2 ] sin cos ) = RHS Prove the following trigonometric identity. tan2 x sec x + 1 cos x cos x sec2 x 1 (sec x 1)(sec x + 1) LHS = = sec x + 1 (sec x + 1) 1 cos x 1 1= = RHS = sec x 1 = cos x cos x 10. Prove the following trigonometric identity. (tan sec )2 LHS = = = 11. 1 1 sin 1 + sin 2 1 sin cos cos (1 sin )2 cos2 (1 sin )2 (1 sin )2 1 sin = = 2 (1 sin )(1 + sin ) 1 + sin 1 sin Find exact value of sin(195o ). sin(195o ) = sin(150 + 45 ) = sin 150 cos 45 + cos 150 sin 45 p p p p p 1 2 3 2 2 6 = = 2 2 2 2 4 12. Find exact value of cos(17 =12). cos(17 =12) = sin(255 ) = sin(210 + 45 ) = sin 210 cos 45 + cos 210 sin 45 p p p p p 1 2 3 2 2 6 = = 2 2 2 2 4 2 = RHS 13. If sin = 1 5 nd the exact values of the following. sin( =4) = sin cos =4 cos sin =4 p " p # p 2 2 1 24 [sin cos ] = = 2 2 5 5 p p 2(1 24) = 10 cos( =3) = cos cos =3 + sin sin =3 p p p 3 24 1 1 3 1 + sin = + = cos 2 2 5 2 5 2 p p ( 24 + 3) = 10 tan + tan =6 tan( + =6) = 1 tan tan =6 = 1 14. p p1 + 3 3 24 p 3 p1 3 24 Find the exact value of sin sin 1 4 5 tan 1 Let y = sin 1 Let z = tan 1 3 4 = sin(y 4 5 1 sin sin 4 5 3 4 tan , sin y = , tan z = 1 3 4 4 =) draw a triangle in quad IV 5 3 =) draw a triangle in quad I 4 z) = sin y cos z cos y sin z 4 4 3 3 = 1 = 5 5 5 5 15. Find the exact value of the expression: sin 105o = sin(60 + 45 ) = sin 60 cos 45 + cos 60 sin 45 p p p p p 3 2 1 2 6+ 2 = + = 2 2 2 2 4 16. Find the exact value of the expression: 5 12 = sin 75 sin = sin(45 + 30 ) = sin 45 cos 30 + cos 45 sin 30 p p 6+ 2 = 4 3 17. For the angles 18. If sin = 4 5 and and shown in the gures, nd cos( + ). Refer to picture given in class. lies in quadrant III, nd sin( =2). sin( =2) r 1 cos = + 2 s r r 3 1 8 4 2 5 = = = =p 2 10 5 5 19. If sin = 3 7 and lies in quadrant II, nd cos( =2). cos( =2) r 1 + cos = + 2 s s s p p 40 1+ 7 7 40 7 = = = 2 14 2 3 20. Find the exact value of sin(2x), if cos x = p 2 10 14 and 0o < x < 90o . sin(2x) = 2 sin px cos x p 5 2 4 5 = 2 = 3 3 9 21. Find the exact value of cos(2x), if tan x = 5 4 and x lies in quadrant III. cos(2x) = cos2 x 4 p = 41 22. sin2 x 2 p 5 41 2 = 16 25 41 = 9 41 Verify the identity: sin 4 4 sin( ) cos( ) cos(2 ) LHS = 2 sin(2 ) cos(2 ) = 2 2 sin cos cos(2 ) = 4 sin( ) cos( ) cos(2 ) = RHS 23. Verify the identity: cos 4x 8 cos4 x LHS = 2 cos2 (2x) 2 = 2[2 cos x 4 = 2[4 cos x 4 = 8 cos x 4 8 cos2 x + 1 1 1]2 1 2 4 cos x + 1] 2 1 8 cos x + 1 = RHS 24. Express the following product as a sum. sin 6 sin 4 OMIT for this test. 25. Express the following product as a sum. cos 3 cos OMIT for this test. 26. Express the following sum as a product. sin 5 sin 3 OMIT for this test. 27. Express the following sum as a product. cos 3 + cos 2 OMIT for this test. 28. Prove the identity: sin 4x + sin 8x sin 4x sin 8x OMIT for this test. 5 tan 6x