Download TRIGONOMETRY[1] - Vasant Valley School

Mathematics
Name:
Class:
TRIGONOMETRY
SECTION – A( 1 mark each)
1. If  = 45°, the value of cosec2  is
1
(A)
2
(B) 1
(C)
1
2
(D) 2
2. Sin (60° +  ) – cos (30° -  ) is equal to
(A) 2 cos 
(B) 2 sin 
(C) 0
(D) 1
(C) -2
(D) 0
(C) cos A
(D) sin A
(C) 2
(D)
3.If sin A + sin2 A = 1, then the value of cos2 A + cos4 4 is
(A) 2
4.
(B) 1
The value of [(sec A + tan A) (1 – sin A)] is equal to
(A) tan2 A
(B) sin2 A
5.In fig., if D is mid-point of BC, the value of
(A)
1
3
(B) 1
tan x
is
tan y
1
2
6.In the given figure, ACB = 90°, BDC = 90°, CD = 4 cm, BD = 3 cm, AC = 12 cm cos A – sin A is
equal to
(A)
5
12
(B)
5
13
(C)
7
12
(D)
7
13
7.The value of tan1°. tan2°. tan3° ………. tan89° is :
(A)0
(B) 1
8.Given that sin A =
(C) 2
(D)
1
2
1
1
and cos B =
then the value of A + B is :
2
2
(A) 30°
(B) 45°
(C) 75°
(D) 15°
(B) -5
(C) 0
(D) 5
(B) 2 sin A
(C) 2 cos A
(D) sec A
9.The value of 5 tan2 - 5 sec2  is :
(A) 1
10.
 cos A

 sin A  is :

 cot A

(A) cot a
11.If cot  =
(A)
1  cos   1  cos   is :
7
then the value of
8
1  sin   1  sin  
49
64
(B)
8
7
(C)
64
49
(C)
4
4
(D)
7
8
12.The value of cos 60° sin 30° + sin 60° cos 30° is :
(A)
1
4
(B)
3
4
(D)
2
4
13.If cos (20 + ) = sin 30°, then the value of  is :
(A) 20°
(B) 50°
(C) 30°
(D) 40°
14.The value of sine  cos ( 90° - ) + cos  sin (90° - ) is :
(A) 1
(B) 0
(C) 2
(D) -1
15.From the figure 1, the value of cosec A + cot A is :
(A)
bc
a
(B)
ab
c
(C)
a
bc
(D)
b
ac
16.If a cos  + b sin  = 4 and a sin  - b cos  = 3, then a2 + b2 is
(A) 7
(B) 12
(C) 25
(D) None
17.If cosec2  (1 + cos ) ( 1 – cos ) =  , then the value of  , then the value of  is
(B) cos2 
(A) 0
18.If sin =
(C) 1
(D) -1
1
and  is acute, then (3 cos - 4cos2) is equal to :
2
(A) 0
(B)
1
2
(C)
1
6
(D) -1
19.If x cos A = 1 and tan A = y, then x2 – y2 is equal to :
(A) tan A
(B) 1
(D) – tan A
(C) 0
20.[Cos4 A – sin4 A] is equal to :
(A) 2 cos2 A + 1 (B) 2 cos2 A – 1 (C) 2 sin2 A – 1 (D) 2 sin2 A + 1
21.The value of the expression [( sec2  - 1) (1 – cosec2 )] is :
(A) -1
(B) 1
1
2
(C) 0
(D)
(C) sec 90°
(D) sin 90°
22.Which of the following is not defined?
(A) cos 0°
23.The value of
(A)
1
2
(B) tan 45°
tan 45
is :
sin30  cos 30
(B) 1
(C)
1
2
(D) 2
24.(sec A + tan A) (1 – sin A) is equal to :
(A) sec A
(B) sin A
(C) cosec A
(D) cos A.
25.If sin 2  = cos ( - 6°) where 2  and ( - 6°) are both acute angles then the value of  is:
(A) 16°
(B) 32°
26.Given that cos  =
(C) 48°
(D) 45°
2 sec 
1
the value of
is :
2
1  tan2 
(A) 1
(B) 2
27.If sec A = cosec B =
(C)
1
2
(D) 0
15
then A + B is equal to :
12
(A) zero
(B) > 90°
(C) 90° (D) < 90°
28.If A is an acute angle in a right  ABC, right angled at B, then the value of sin A + cos A is :
(A) equal to one
(B) greater than one
(C) less than one
(D) equal to two
29.If x = 2 sin2  , y = 2 cos2  + 1 then the value of x + y is :
(A) 2
(B) 3
(C)
1
2
(D) 1
30.If cos (+) = 0, then sin ( - ) can be reduced to :
(A) cos 
31.If cosec  =
(A) 3
(B) cos 2 
(C) sin 
(D) sin 2 
3
, then 2 (cosec2 + cot2) is :
2
(B) 7
(C) 9
(D) 5
(C) 1
(D) 0
32.If sin  + sin2  = 1, the value of (cos2  + cos4 ) is :
(A) 3
(B) 2
33.If sin  = cosec  = 5 , then the value of sin  + cosec  is :
(a) 3
34.If cos 3 =
(b) 1
3
; 0 <  < 20° then the value of  is :
2
(c) 3
(d) 9
(a) 15°
(b) 10°
(c) 0°
(d) 12°
35.If 3 cos  = 1, then the value of cosec  is :
(a) 2 2
3
(c)
2 2
2 2
3
(d)
4
3 2
1
, then the value of sin  (sin  - cosec ) is
2
36.If sin  =
(A)
(b)
3
4
(B)
3
4
(C)
3
2
(D)
 3
2
37.The value of sin2 30° + cos2 45° + cos2 30° is :
(A)
1
2
(B)
3
2
(C)
3
2
(D)
2
3
38.Expressing sin A in terms of cot A is
(A)
(C)
1  cot2 A
cot A
1
1  cot2 A
(B)
1  cot2 A
cot A
(D)
1  cot2 A
cot A
39.If tan 2A = cot (A – 18°), then the value of A is
(A) 18°
(B) 36°
(C) 24°
(D) 27°
40.Value of (1 + tan  + sec ) (1 + cot  - cosec ) is :
(A) 1
(B) -1
(C) 2
(D) -4
(C) 2
(D) -1
(C) cosec 90°
(D) sec 90°
41.The value of [ sin2 20° + sin2 70° - tan2 45° ] is :
(A) 0
(B) 1
42.Which of the following is defined ?
(A) tan 90°
43.If sin (A – B) =
(A) 45°
(B) cot 0°
1
1
and cos (A + B) = , then the value of B is :
2
2
(B) 60°
44.If sin  = cos , then value of  is :
(C) 15°
(D) 0°
(A) 0°
45.If cot A =
(A)
(B) 45°
(C) 30°
(D) 90°
12
, then the value of (sin A + cos A)  cosec A is :
5
13
5
(B)
17
5
(C)
14
5
(D) 1
46.9 sec2  - 9 tan2  is equal to :
(A) 1
(B) -1
(C) 9
(D) -9
1
2
(D) -1
47.cos 1°, cos 2°, cos 3° ……. Cos 180° is equal to :
(A) 1
(B) 0
48.The maximum value of
(A) 0
48.If tan A =
(A)
(C)
1
is :
cos ec 
(B) -1
(C) 1
(D)
3
2
3
and A + B = 90° then the value of cot B is equal to :
4
4
3
(B)
1
2
(C)
3
4
(D) 1
49.If x, tan 45°. cot 60° = sin 30°. cosec 60°, then the value of x is :
(A) 1
(B)
1
4
(C)
1
2
(D) 3
50.If sin 3 = cos ( - 26°) where 3 and (0 – 26°) are acute angles, then value of  is :
(A) 30°
(B) 29°
(C) 27°
(D) 26°
51.If sin  = cos , then the value of cosec  is :
(A) 2
52.If sec A = cosec B =
(A) zero
(B) 1
(C)
2
3
(D) 2
12
then A + B is equal to :
7
(B) 90°
(C) < 90°
(D) > 90°
SECTION – B( 2 mark each)
1.If sec 4 A = cosec (A – 20°) where 4 A is an acute angle, find the value of A.
2.cot  =
1  sin   1  sin  
7
, find the value of
8
1  cos   1  cos  
3.If 5 tan  = 4, find the value of
5 sin   3 cos 
.
5 sin   2 cos 
4.Prove that
sin 
cos 

sin  90   
cos  90  
5.Prove that
cos 
cos 
= 2 sec .
1  sin  1  sin 
6.Evaluate :
cos ec2 90     tan2 

2
2
4 cos 48  cos 42



= sec . cosec 
2 tan2 30 sec2 52 sin2 38
cos ec2 70  tan2 20
7.In figure, ABC is right triangle, D is mid point of BC.
Show that
tan 
1

tan 
2
8.In PQR right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Find the value of sin P.
9.If sin (A-B) =
10.If  and
1
1
, cos (A+B) = and 0 < A + B < 90° and A > B then find the values of A and B
2
2
1
are the zeros of the polynomial 4x2 – 2x + (k – 4), find the value of k.

11.If 3 cot A = 4, find the value of
cos ec2 A  1
cos ec2 A  1
12.If tan  =
1
7
, find the value of
cos ec2   sec2 
.
cos ec2   sec2 
13.If A,B,C are interior angles of  ABC, show that :
B  C 
2 A
=1
 - tan
2

ccosec2 
 2
14.sec2 + cot2 (90 - ) = 2 cosec2 (90 - ) – 1.
15.Find the value of k, if
cos 20
2 cos 
k

 .
sin70
sin  90   
2
16.If sin  + cos  = m and sec  + cosec  = n, then prove that n (m2 – 1) = 2m.
17.Prove that : 1 +
cot2 
1

1  cos ec 
sin 
18.If 3 tan  = 3 sin , then prove that sin2  - cos2  =
1
.
3
19.If 7 sin2  + 3 cos2  = 4, then prove that sec  + cosec  = 2 +
20.If tan (A+B) = 3 and tan (A-B) =
1
3
2
3
.
, 0° < A+B  90°: A> B, find A and B.
21.If sin 3 A = cos (A – 26) where 3 A is an acute angle, find the value of A.
22.In ABC, right angled at B, AB = 5 cm, ACB = 30°. Find BC and AC.
23.If sin  + sin2 = 1, then find the value of cos2  + cos4 .
24.If sin (A + B) = cos (A –B) =
25.If tan  =
3
and A, B (A > B) are acute angles, find the values of A and B.
2
5
find the value of :
12
cos   sin 
cos   sin 
SECTION – C( 3 mark each)
1.Prove that
tan   cot 
= tan2  - cot2 
sin  cos 
2.In  ABC is right angled at B, BC= 7 cm and AC– AB = 1cm. Find the value of cos A – sin A.
B  C 
A
2
3.If A,B,C are interior angles of ABC, show that Sec2 
 -1 = cot
2
2


4.Prove that :
cos(90  )
= 2 cosec 
1  sin(90  )
5.If cot  =
4 sin   3 cos 
4
, evaluate
.
3
4 sin   3 cos 
6.If tan  =
24
, find the value of sin  + cos .
7
7.Evaluate :
cos ec2 90     tan2 

2
2
5 cos 48  cos 42


2
1
sin 48° sec 42° - tan2 60°.
5
5
8.Prove that (1 + cot  - cosec ) (1 + tan  + sec ) = 2.
9.Prove that
1  sec A
sin2 A

.
sec A
1  cos A
10.If A + B = 90°, then prove that
tan A tanB  tan A cot B
sin2 B
= tan A

sin A sec B
cos2 A
11.Prove that 2 sec2  - sec4  - 2 cosec2  + cosec4  = cot2  - tan4 
12.Prove that
13.If
tan 
cot 
= 1 + tan  + cot .

1  cot  1  tan 
x2
y2
1
y
x
y
cos  
sin  = 1 and sin  
cps  = 1, prove that 2  2 = 2
a
b
a
b
a
b
C
A B
  sec 2 .

14.If A, B, C are interior angles of ABC, prove that cosec 
 2
15.If sin (2A + 45°) = cos (30° - A), Find A.
16.Prove that
sec   1

sec   1
sec   1
= 2 cosec .
sec   1
17.Prove that
1  sin A
= sec A + tan A
1  sin A
2 sin68 2 cot15 3 tan 45 tan20 tan 40 tan50 tan70


cos 22 5 tan75
5
18.Evaluate :
19.Prove that sec  (1 – sin ) (sec  + tan ) = 1.
20.If sec  = x +
1
1
then prove that sec  + tan  = 2x or
.
4x
2x
21.If A, B, C are interior angles of  ABC, show that :
22.Prove that :
23.Prove that :
24.If sin  =
25.Prove that

cot 90o  
tan 



cos ec 90o  

o
tan 90  


Cos2
A
B  C 
+ cos2 
 = 1.
2
 2 
= sec2  .
tan 
cot 
= 1 + sec . cosec 

1  cot  1  tan 
tan   4
m
, then find the value of
.
n
4 cot   1
(cosec A – sin A) (sec A – cos A) =
1
tan A  cot A
26.If tan A = n tan B sin A = m sin B, prove that cos2 A =
m2  1
.
n2  1
27.Evaluate :
Sin (50° + ) – cos (40° - ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89° + sec (90 - ) cosec  tan (90 - ). cot .
28.Prove that (sin  + cosec )2 + (cos  + sec )2 = 7 + tan2  + cot2 
29.Prove that (cosec  - sin ) ( sec  - cos ) =
1
tan   cot 
30.If sin  + cos  = p and sec  + cosec  = q, show that q(p2 – 1) = 2p
31.If sin  + cos  = 2 sin (90 - ), then find the value of tan .
sin39
 2 tan 11° tan 31° tan 45° tan 59° tan 79° - 3 (sin2 21° + sin2 69°)
cos 51
32.Evaluate :
33.Prove that (cosec  - cot )2 =
1  cos 
1  cos 
34.If m sin  + n cos  = p and m cos  - n sin  = q then prove that m2 + n2 = p2 + q2
SECTION – D(4 marks each)
1.Without using trigonometric tables, evaluate the following :
sec 37o
+ 2 cot 15° cot 25° cot 45° cot 75° cot 65° - 3 (sin218° + sin272°)
cos ec 53o
tan 
cot 
= 1 + sec  cosec  .

1  cot  1  tan 
2.Prove that:
3.If 2 cos  - sin  = x and cos  - 3 sin  = y. Prove that 2x2 + y2 – 2xy = 5.
4.Prove that
cot   1  cos ec
1

cot   1  cos ec
cos ec  cot 
5.If tan  + sin  = m and tan  - sin  = n, show that (m2 – n2)2 = 16 mn
1  sinA
= sec A + tan A
1  sin A
6.Prove that
7.Prove that
cot A  cos ec a  1
1  cos A

cot A  cos ec A  1
sin A
8.Prove that
sin A  cos A
sin A  cos A
2


.
sin A  cos A
sin A  cos A
sin2 A  cos2 A
9.Prove that
cos A
1  sin A
= 2 sec A.

1  sin A
cos A
10.If tan  + sin  = m and tan  - sin  = n. Show that m2 – n2 = 4 mn
11.Prove that
tan 
cot 
= 1 + tan  + cot 

1  cot  1  tan 
3 tan35 tan 40 tan50 tan55 
12.Without using trigonomietric tables evaluate :
13.If tan A = 2. Evaluate sec A sin A + tan2 A – cosec A

4 cos2 39  cos2 51

1
tan2 60
2
14.If tan A = 2 - 1. Show that sin A cos A =
2
4
15.Prove that sin6  9 cos6  = 1 – 3 sin2  cos2 .
16.In ABC, B = 90 AB = 3 cm and BC = 4 cm. Find
(i) sin C
(ii) cos C
1
17.If tan A =
3
(iii) sec A
(iv) cosec A
, ABC is right angled at B.
18.Find the value of sin A cos C + cos A sin C.
19.Prove that :
tan 
cot 
= 1 + sec . Cosec 

1  cot 
1  tan 
20.Prove that :
1
1
1
1



cos ec A  cot A
sin A
sin A
cos ec A  cot A
21.Prove that : sec2  -
sin2   2 sin4 
= 1.
2 cos4   cos 2
22.If sec  - tan  = 4 then prove that cos  =
8
.
17
23.Find the value of sin2 5° + sin2 10° + sin2 80° + sin2 85°.
24.Prove that
25.Prove that
1  sec A
sin2 A

.
sec A
1  cos A
tan   1  sec 
1

tan   1  sec 
sec   tan 
26.If tan  + sin  = m and tan  - sin  = n, show that m2 – n2 = 4 mn .
27.Prove that
cos ec A  1
= sec A + tan A
cos ec A  1
28.Prove that
cos   sin   1
= cosec  + cot .
cos   sin   1
29.If x = r sin A cos C, y = r sin A sin C, z = r cos A, prove that r2 = x2 + y2 + z2.
30.Prove that
1  cos A
= cosec A + cot A.
1  cos A
31.Prove that :
32.Evaluate
sin   cos 
sin   cos 
sin   cos 
2

.
sin   cos 
2 sin2  1
sin35
cos 55 .cos ec 35o

cos 55
tan5 tan25 tan 45 tan65 tan85
33.If a sin  + b cos  = c, then prove that a cos  - b sin  = a2  b2  c2 .
34.Prove that
1+cos θ+sinθ
1+sinθ
.
=
1+cos θ- sinθ
cos θ
35.Without using trigonometric tables, evaluate
cosec2  90°- θ - tan2 θ 2 tan2 30° sec2 52° sin2 38°
.
tan2 20°- cosec2 70°
4 cos2 48°+cos2 42°


 1  sin A 
cot2 A (sec A  1)
 sec2 A 

1  sin A
 1  sec A 
36.Prove that
37.Prove that
1  cos   sin 
= cosec  + cot .
cos   1  sin 
sec2 (90  )  cot2 2 cos2 60 tan2 28 tan2 62
cot 40

tan50
2(sin2 25  sin2 65)
3(sec2 43  cot2 47)
32.Evaluate :
33.Show that
sin   2 sin3 
= tan 
2 cos2   cos 
34.Prove that sin A (1+tan A) + cos A (1+cot A) = sec A + cosec A.
35.Prove that
tan   sec   1
1  sin 

tan   sec   1
cos 
36.Is sec  + tan  = p, show that
37.If sec  = x +
p2  1
= sin .
p2  1
1
1
, then prove that tan  + sec  = 2x or
.
4x
2x
38.Prove that sec4  - tan4  + 1 + 2 tan2 .
39.Find the value of
cos 20
cos 70
- 8 sin2 30°

sin70
sin20
40.Prove that tan2  + cot2  + 2 = sec2  cosec2 