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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  
 30
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

; nZ
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


; nZ
(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
; nZ
6

3
2m

or  2 p  1
5
2