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ASSIGNMENT CLASS XI TRIGONOMETRY Q1. Find the degree measure corresponding to the following radian measures: 2 (a) 15 c (b) 8 c 9 (c) 5 c 5 (d) 6 c Q2. Find the radian measures corresponding to the following degree measures: (a) 3400 (b) 750 (c) 37 030' (d) 5200 Q3. Find the magnitude, in radians and degrees, of the interior angle of a regular: (a) pentagon (b) octagon (c) heptagon (d) duodecagon Q4. In a right angled triangle , the difference between two acute angles is . Express the angles in 18 degrees. Q5. The angles of a triangle are in A.P. such that the greatest is 5 times the least. Find the angles in radians. Q6. The angles of a triangle are in A.P. and the number of degrees in the least to the number of radians in the greatest is 60: . Find the angles in radians. Q7. The number of sides of two regular polygons are as 5:4 and the difference between their angles is 90 . Find the number of sides of the polygons. Q8. The perimeter of a certain sector of a circle is equal to half that of the circle of which it is a sector. Find the circular measure of the angle of the sector. Q9. The larger hand of a big clock is 35 cm long. How many cm does its tip move in 9 minutes? Q10. (a) Find the angle between the minute and hour hands of a clock at 8:30 . (b) Find the angle between the minute and hour hands of a clock at 3:40. Q11. If cot 12 and lies in the second quadrant, find the values of other five trigonometric 5 functions. Q12. Find the values of the following trigonometric ratios: 5 (a) sin 3 15 (f) cot 4 (b) sin 30600 (g) cos 5700 (c) tan 11 6 (h) sin 3300 (d) cos 11250 (i) cos ec 12000 (e) tan 3150 (j) tan 5850 Q13. Prove that: 3sin 5 sec 4sin cot 1 6 3 6 4 Q14. Evaluate the following: (a) sin 7 7 cos cos sin 12 4 12 4 (b) cos 2 2 cos sin sin 3 4 3 4 1 5 3 Q15. If cot , , and sec , , , find the value of tan . 2 2 3 2 4 12 3 3 Q16. If cos x , cos y ; x 2 and y 2 , find cos x y and sin x y . 5 13 2 2 Q17. Prove that 3 5 7 (i) cos 2 cos 2 cos 2 cos 2 2 8 8 8 8 3 5 7 (ii) sin 2 sin 2 sin 2 sin 2 2 4 4 4 4 Q18. a b (a) Prove that the equation 2 4ab sin 2 is possible only when a b . (b) Is the equation 2sin 2 cos 4 0 possible? (c) Prove that sec 2 cos ec 2 4 1 (d) Show that cos x is impossible, if x is real. x 1 (e) Find the value of cos for which 2cos a is possible, where a R a Q19. (a) Find the sign of the expression: sin1000 cos1000 . 3 sin cos as a single term consisting: (b) Reduce (i) Sine only (ii) Cosine only Q20. Prove that: (a) (b) cos ec cos 270 tan 180 cos cos 900 sec 2700 sin 1800 0 0 sin 360 cos 360 cos ec sin 270 1 sin 1800 cos 900 tan 2700 cot 3600 0 Q21. If cos x cos y Q22. If tan x 0 0 1 1 x y 3 and sin x sin y , prove that tan . 3 4 2 4 1 1 and tan y , show that cos 2 x sin 4 y . 7 3 Q23. (a) If tan A m 1 , then prove that A B . and tan B 4 m 1 2m 1 (b) If tan m 1 , show that . and tan 4 m 1 2m 1 Q24. Prove the following: (a) sin 2 cot 1 cos 2 (b) 1 sin 2 cos 2 cot 1 sin 2 cos 2 (c) 1 sin cos tan 1 sin cos 2 (d) cos tan 1 sin 4 2 (e) 1 sin x x tan 1 sin x 4 2 x x (f) tan tan 2 sec x 4 2 4 2 Q25. If sec tan 4 , find the values of sin , cos , sec and tan . Q26. Find the quadrant in which lies if sin 1 and tan 1 2 Q27. Find a pair of values of R and from R cos 3 and R sin 1 Q28. Find all angles between 00 and 3600 satisfying 3tan 2 x 1 Q29. If sin cos 0 and lies in IV quadrant, find sin and cos . Q30. If tan x cot x 2 , prove that tan n x cot n x 2 ; n N Q31. (a) Prove that: cos sin 2 cos . 4 Q32. If sin A (b) Prove that: 3 cos ec 20 0 sec 200 4 1 3 and cos B , prove that A B . 4 5 10 Q33. Prove that 2 2 2cos 4 2cos Q34. Find maximum and minimum values of sin cos . Q35. If tan x tan y a and cot x cot y b , prove that 1 1 cot x y . a b Q36. Prove that: (a) 4sin A sin 600 A sin 600 A sin 3 A (c) sin100 sin 500 sin 600 sin 70 0 3 16 (b) cos A cos 1200 A cos 1200 A 0 (d) cos 200 cos 400 cos800 1 8 Q37. Prove that: 1 (a) cos cos cos cos 3 3 3 4 2 1 (b) sin sin sin sin 3 3 3 4 Q38. Prove that: (a) cos cos cos cos 4cos cos8 A cos 5 A cos12 A cos 9 A tan 4 A sin 8 A cos 5 A cos12 A sin 9 A (c) (d) sin 3 A sin 5 A sin 7 A sin 9 A tan 6 A (e) (b) cos 3 A cos 5 A cos 7 A cos 9 A (f) sin 8 cos sin 6 cos 3 tan 2 cos 2 cos sin 3 sin 4 Q39. If sin A Q40. cos cos 2 2 2 sin11A sin A sin 7 A sin 3 A tan 8 A cos11A sin A cos 7 A sin 3 A (g) sin 5 A sin 7 A sin 8 A sin 4 A cot 6 A cos 4 A cos 7 A cos 5 A cos8 A sin A B 2sin A sin A B tan A cos A B 2 cos A cos A B 3 , where 00 A 900 , find the values of sin 2 A , cos 2 A , tan 2 A and sin 4 A . 5 (a) Find the greatest value of sin x cos x . (b) If x y 900 , find the maximum and minimum values of sin x sin y . Q41. Find the value of sin and cos . 8 8 Q42. Find the most general value of satisfying the equations: sin 1 1 and tan . 2 3 Q43. Find general solution of: (a) cos 3 1 2 (c) tan tan 2 1 (b) tan 3 x 1 Q44. Solve the following equations: (a) 2 tan 2 x sec2 x 2 ; 0 x 2 3 (c) cot 2 30 sin (b) sin 2 sin 4 sin 6 0 (e) tan tan 2 tan tan 2 1 (f) (g) cot cos ec 3 (h) sin 2 cos (i) 2sin 2 x 3 cos x 1 0 (j) cos cos 2 cos 3 0 (d) tan 2 1 3 tan 3 0 3 cos sin 2 1 4 Q45. Solve the following trigonometric equations: (a)sin 3x cos 2 x 0 (c) 2 tan x cot x 1 0 (b)3tan x cot x 5cos ecx (d) cos 3x cos x cos 4 x cos 2 x Q46. (a)What is the minimum value of 3cos x 4sin x 8 . (b) If sin x cos x 1 , what is the value of sin 2x . (i )3sin x Q47. Draw the graph of: (ii ) sin 2 x (iii) cos x (iv)3cos 2 x Q48. The angles of a ABC are in A.P. and it is being given that b : c 3 : 2 , find A . Q49. In ABC , if a 3, b 5, c 7 , find cos A , cos B , cos C . 6 square units. 2 Q50. In ABC , if a 2 , b 3 , c 5 , show that its area is Answers (b) 22030' 1. (a) 240 17 2. (a) 9 c c 5 (b) 12 c , cos 10. (a) 75 14. (a) 13 (b) 18 12 13 ,sec 13 12 (e) -1 (b) 19. (a) positive 25. sin 11. cos ec 3 2 (b) 0 3 2 (h) 1 2 (i) 15. 2 11 16. (b)(i) 2sin 6 40. (a) 13 5 5 , tan ,sin 5 12 13 (c) 2 3 1 3 (d) 1 2 (j) -1 33 16 , 65 65 18. (e) a 1 (ii) 2 cos 3 15 8 17 15 , cos , sec , tan 17 17 8 8 24 7 24 336 , , , 25 25 7 625 c c c 3 1 2 2 28. x 300 ,1500 , 2100 and 3300 39. 26 (d) 9 c 12. (a) (g) (f) 1 3 2 c 3 5 5 0 ' '' 0 (b) ; 1350 (c) ; 128 3417 (d) ; 150 4 7 6 5 5. , , 6. , , 7. 10 , 8 8. 2 radians 9 3 9 6 3 2 0 9. 33 cm (d) 1500 5 (c) 24 c 3 3. (a) ; 1080 5 4. 500 , 400 (c) 3240 29. 1 1 , 2 2 1 2 (b) 1 1 and 2 2 26. Third 27. R 2, 6 34. Maximum : 2 and minimum : 2 41. 2 1 , 2 2 2 1 2 2 42. 2n 11 , n is any integer 6 43. (a) 1 or n where tan 4 2 n (b) or m where m , n Z 4 3 (c) n (d) n 2n 2 ; nZ 3 9 5 7 11 , , , 6 6 6 6 n 1 m 1 (c) n 1 or m 1 ; m , n Z 6 2 n 5 or m ; m , n Z (e) ; n Z (f) 2n or 2n ; n Z 4 3 3 12 12 12 ; nZ (h) 2n ; n Z 3 3 2 (j) 2n 1 or 2n ; n Z 4 3 45. (a) x 2n (c) x n 2n or x 2 5 10 3 1 or x m , tan 4 2 (d) x 2n , (b) 0 (i) (ii) (iii) (iv) 49. 13 11 1 , , 14 14 2 (i) 2n (b) 2n 47. 48. 750 n ; n Z 3 12 44. (a) (g) 2n 46. (a)3 (b) x 5 ; nZ 6 3 2m or 2 p 1 5 2