TRIGNOMETRY RAKESH YADAV Practice Set https://www.amazon ...
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TRIGNOMETRY RAKESH YADAV Practice Set
MATHS OF WIZARD
1 ? tan2 90? ?
1. What is the value of
? 1?
cos2 90? ?
gSa\
(a) 0 (b) 1
1 ? tan2 90? ? cos2 90? ? ? 1 dk eku Kkr djsa\
(a) ? sin 2 (b) ? cos 2 (c) cos 2
(c) ?1
(d)
2
6. What is the value of
tan5 + tan3 = ? 4cos4 tan5 ? tan3
(d) sin 2
1 ? tan2221?
2. Find the value of
2?
1 + tan2221?
2
1 ? tan2221?
2 dk eku Kkr djsa\
1 + tan2221? 2
tan5 + tan3 4cos4 tan5 ? tan3
dk eku D;k gSa\
(a) sin 2 (b) cos 2 (c) tan 4 (d) cot 2
(a) 2
1 (b)
2
1 (c) ?
2
3 (d) ?
2
3. What is the value of cos15??cos165?=?
cos15??cos165? dk eku D;k gSa\
3 (a)
2
2 (b)
3 ?1
3 1 (c)
2
3 1 (d)
2
sin7x ? sin5x cos6x ? cos4x
4.
?
?
cos7x + cos5x sin6x + sin4x
(a) 2tanx
(b) tanx
(c) 2sinx
(d) 2cosx
5. What is the value of
sinx + siny sinx ? siny = ? cosx + cosy cosy ? cosx
sinx + siny sinx ? siny cosx + cosy cosy ? cosx
dk eku D;k
1
7. What is the value of sin(B?C)cos(A? D)+sin(A?B)cos(C?D)+sin(C?A)cos(B?D)?
sin(B?C)cos(A?D)+sin(A?B)cos(C?D)
+sin(C?A)cos(B?D) dk eku D;k gSa\
3 (a)
2
(b) ?3 (c) 1 (d) 0 8. What is the value of
180? ? 180? ? 9 180? ? 13
cos
cos
sin
2 2
2
180? ? 180? ? 9 180? ? 13
cos
cos
sin
2 2
2
dk eku D;k gSa\
(a) sin 2sin 4 (b) cos 2 cos 6
(c) sin 2sin 6
(d) cos 2 cos 4
9. What is the value of [(sin7x?sin5x)? (cos7x+cos5x)]?[(cos6x?cos4x)?(sin6x +sin4x)]?
[(sin7x?sin5x)4? (cos7x+cos5x)]?[(cos6x?
cos4x)?(sin6x +sin4x)] dk eku D;k gSa\
(a) 1
(b) 2tanx
(c) tan2x
? cos + 2cos3 + cos5 + sin2tan3
3x
(d)
tan
2
10. What is the value of [(cos7A+cos5A)? (sin7A?sin5A)]?
dk eku D;k gSa\
(a) cos 2 (b) sin 2
[(cos7A+cos5A)? (sin7A?sin5A)] dk eku D;k gSa\
(c) tan 2 (d) cot sin 2
(a) tanA (b) tan4A (c) cot4A
13. What is the value of [sin(90?+2A)(4? cos2(90??2A)]?
[sin(90?+2A)(4?cos2(90??2A)] dk eku D;k gSa\
(d) cotA
1 ? sin 90? ? 2A 11. What is the value of 1 + sin 90? + 2A ?
(a) 2(cos2A?sin3A) (b) 2(cos3A+sin3A) (c) 4(cos6A+sin6A) (d) 4(cos6A?sin6A)
1 ? sin 90? ? 2A 1 + sin 90? + 2A
dk eku D;k gSa\
(a) sinAcosA (b) cot2A (c) tan2A (d) sin2AcosA
12. What is the value of
cos3 + 2cos5 + cos7
? cos + 2cos3 + cos5 + sin2tan3
cos3 + 2cos5 + cos7
14. What is the value of
sin x + y ? 2sinx + sin x ? y
cos
x
?
y
+
cos
x
+
y
?
2cosx
?
sin10x cos10x
? +
sin8x cos8x
?
sin x + y ? 2sinx + sin x ? y
cos x ? y + cos x + y ? 2cosx
? sin10x ? sin8x cos10x + cos8x
2
dk eku D;k gSa\
(a) 0 (c) 1
(b) tan2x (d) 2tanx
sin2x
15.
The value of
x sin
4
sin2x
x sin
dk eku Kkr djsa\
4
xx (a) 4 cos .cos .cos x
42
xx (b) 4 cos .cos .cos x
22
xx (c) 8 cos .cos .cos x
42
xx (d) cos .cos .cos x
22 16. The value 3sin20??4sin320?
3sin20??4sin320? dk eku Kkr djsa\
(a) 3
(b) 2
1 (c)
2
3 (d)
2
2cosy ? 1 17. If cosx = 2 ? cosy , then find the value
xy of tan cot
22
;fn
2cosy ? 1 cosx =
gS] rkstan x cot y
2 ? cosy
22
dk eku Kkr djsa\
(a) 3
(b) 2
3 (c)
2
1 (d)
3
3
SOLUTION
1. (a)
1 ? tan2 90 ? ? 1 cos2 90 ?
By putting 45
2 cos 6x sin x 2 sin 5x ? sin x
?
2 cos 6x cos x 2 sin 5x cos x
= tanx?(?tanx)=tax+tanx
= 2tanx
5. (b)
1 ? tan2 45 ? 1 cos2 45
0?1 ?1
By satisfying in option (a)
? sin 2 ?sin90? ?1 2. (b)
As we know-
sinx + siny sinx ? siny cosx + cosy cosy ? cosx
put x = 90?, y = 0
sin 90 sin 0 sin 90 ? sin 0 = cos 90 cos 0 cos 0 ? cos 90
=
1 0 1 ? 0 1 0 1 1 ? 0
6. (b)
1 ? tan2 A cos 2A
1 tan2 A
tan5 + tan3 4cos4 tan5 ? tan3
1
cos
2
22
2
cos
45
1
2
3. (c) cos15??cos165? = cos15??cos(80?15?) = cos15?+cos15? = 2cos15?
Put 60
tan 5 tan180
=
4
cos
240
tan
5
?
tan180
tan 5 =
4 cos 240 tan 5
Q tan180 0
1
1
4 cos 240 4 cos 180 60
3 1 3 1
2
2
2
2
[cos15?=cos(45??30?)] = cos45?cos30?+sin45?sin30?
1 3 11
2 2 22
3 1
22
4. (a)
AB A?B
sin A ? sin B 2 cos
.sin
2
2
AB A?B
and sin A sin B 2sin
. cos
2
2
1
1
?
4
?
1 2
2
from option (b)
cos 2 cos120
= cos(90+30?) Q cos 90 ? sin
?1 = ?sin30?
2 7. (d)
sin(B?C)cos(A?D)+sin(A?B)cos(C?D) +sin(C?A)cos(B?D) Put A=B=C=D=0 sin0?cos0?+sin0?cos0?+sin0?cos0? = 0+0+0=0 8. (b)
AB B?A
and cos A ? cos B 2sin
.sin
2
2
180 ? 180 ? 9
cos
.cos
2 2
AB A?B
and cos A cos B 2 cos
. cos
2
2
180 ? 3 180 ? 13
sin
3
.sin
2
4
9
=
cos
90
?
2
. cos
90
?
2
3 13
sin 90 ?
2
.sin 90 ?
2
9
3 13
= sin .sin cos .cos
22
2
2
1 9
3 13
=
2 2 sin 2 .sin 2
2 cos .cos 2
2
= 1 cos 4 ? cos 5 cos 5 cos 8
2
= 1 2 cos 60.cos 2 cos 2.cos 6
2 9. (b)
sin 7x ? sin 5x cos 6x ? cos 4x ?
cos 7x cos 5x sin 6x sin 4x
2
cos
7x
2
5x
.sin
7x
? 2
5x
=
2
cos
7x
2
5x
. cos
7x
? 2
5x
?
2 sin
6x
2
4x
. sin
4x
2
6x
6x 4x 6x 4x
2sin
2
.cos
2
sin x sin ?x
=
?
cos x cos x
= tanx+tanx
= 2tanx
10. (d)
(cos7A+cos5A)? (sin7A?sin5A)
7A 5A 7A ? 5A
2 cos
2
cos
2
=
7A 5A 7A ? 5A
2 cos
2
sin
2
cos 6A cos A
=
cot A
cos 6A sin A
11. (c)
12. (a)
cos 3 2 cos 5 cos 7 sin 2.tan 3
cos 2 cos 3 cos 5
2 cos 5 cos 2 2 cos 5
=
sin 2 tan 3
2 cos 3 cos 2 cos 3
2 cos 5 cos 2 1 = 2 cos 3 cos 2 1 sin 2 tan 3
cos 5 sin 2sin 3
=
cos 3 cos 3
cos 5 sin 2.sin 3 =
cos 3
2 cos 5 2sin 3.sin 2 =
2 cos 3
2sinA.sinB=cos(A?B)?cos(A+B)
2 cos 5 cos ? cos 5 =
2 cos 3
cos 5 cos =
2 cos 3
2 cos 3.cos 2 =
2 cos 3
= cos 2 13. (d)
sin(90+2A)[4?cos2(90?2A)] = cos2A(4?sin22A) = put value of A = 90 = cos180(4?sin2180) = (?1)(4?0) = ?4 14. (b)
sin x + y ? 2sinx + sin x ? y
cos
x
?
y
+
cos
x
+
y
?
2cosx
? sin10x ? sin8x cos10x + cos8x
2sin x cos y ? 2sin x 2 cos 9x.sin x
=
2
cos
x
cos
y
?
2
cos
x
2 cos 9x cos x
1 ? sin 90? ? 2A 1 + sin 90? + 2A
1 ? cos 2A 1 ? 1 2 sin2 A
=
1 cos 2A
1 2 cos2 A
?1
=
2 sin2 A 2 cos2 A
tan2 A
Q cos 2A 1 ? 2sin2 A 2 cos2 A ? 1
2sin x cos y ? 1
=
2
cos
x
cos
y
?
1
tan
x
tanxtax=tan2x
15. (c)
sin 2x x
sin 4
First we solve sin2x
5
sin2x=2sinx.cosx
=
2 sin
2
x 2
. cos
x
xx = 4sin .cos .cos x
22
=
x x
4
sin
2
2
2
.
cos
2
.
cos
x
xxx = 4 2sin .cos .cos .cos x
442
xxx
4 2sin .cos .cos .cos x
442
=
x
sin
4
xx = 8 cos .cos .cos x
42 16. (d)
3si n 20??4si n 320?=sin3?20? = sin60?
3 =
2
17. (a)
cos x 2 cos y ? 1
1 2 ? cos y
Applying componendo dividendo rule
cos x 1
cos x ? 1
2 cos y ? 1 2 ? cos y 2 cos y ? 1 ? 2 cos y
cos x 1 1 cos y
cos x 1 3 cos y ? 1
2 cos2 x ? 1 1 2
1 ? sin2 x ? 1 2
1 2 cos2 y ? 1 2
3 1
?
2 sin2
y 2
? 1
2 cos2 x
2 cos2 y
y
2
?2 sin2 x 2
3
?2 sin2
y 2
cot x 1 cot2 y 23 2
3 tan2 x .cot2 y
2
2
xy Hence, tan .cot 3
22
6
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