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TRIGONOMETRY
[ BASIC, EQUATIONS, INVERSE, SOLUTION OF Δ ,HT & DISTANCES]
By:- Nishant Gupta
For any help contact:
9953168795, 9268789880
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TRIGONOMETRY
[ BASIC, EQUATIONS, INVERSE, SOLUTION OF Δ ,HT & DISTANCES]
By:- Nishant Gupta
For any help contact:
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
BASICS
Some Formulae
Sin 3 A A 3
ASin 3
SinASin (^)
Cos 3 A A 3
A Cos 3
CosACos (^)
- A Tan 3 A 3
ATan 3
TanATan (^)
- tan+ tan
3 tan 3 3
tan 3
5. A 3 / 2
A Sin 3
Sin A Sin 2 2 2
6. Sin( A B)SinA B SinA Sin B
2 2
7. Cos( A B)Cos A B CosA SinB
2 2
2 SinA
Sin 2 A CosACos 2 ACos 4 A...........nterms n
n
n 2
2 n 1
n ........cos 2 n 1
cos 2 n 1
cos 2 n 1
cos
- In a triangle ABC
TanATanBTanC TanATanBTanC &
C
Cot 2
B
Cot 2
A
Cot 2
C
Cot 2
B
Cot 2
A
Cot
- Sin + Sin( + ) + Sin( + 2) + ……+ Sin( + n 1 ) =
Sin
n }Sin 2
n 1 Sin{
TRIGONOMETRY
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
- sin-1(–x) = – sin-1x cosec-1^ (–x)= – cosec-1x sec-1^ (–x) = π – sec-1x
tan-1(–x) = – tan-1x cot-1^ (–x) = π – cot-1x cos-1(–x) = π – cos-1x
- tan-1x + tan-1y = tan- 1 xy
x y
(^) , x y < 1 = π + tan- 1 xy
x y
(^) , x y > 1, x > 0
= - π + tan- 1 xy
x y
(^) , x y > 1,x < 0
= π / 2 , x y = 1 , x > 0 = - π / 2 , x y = 1 , x < 0
- tan-1x - tan-1y = tan- 1 xy
x y
(^) , x y > -1 = π + tan- 1 xy
x y
(^) , x y < -1 , x > 0
= - π + tan- 1 xy
x y
(^) , x y < -1 , x < 0
= π/2 , x y = -1 , x > 0 = - π / 2 , x y = -1 , x < 0
- sin-1x + sin-1y = sin-
2 2 (^) x 1 y y 1 x 0 or (x y > 0 & x (^2) +y (^2) 1)
= π – sin-
2 2 (^) x 1 y y 1 x if x & y > 0 & x (^2) +y (^2) > 1
= - π – sin
-
2 2 x 1 y y 1 x if x & y < 0 & x^2 +y^2 >
- sin-1x- sin-1y =sin-
2 2 (^) x 1 y y 1 x if xy 0 or (x y) > 0 & x (^2) +y (^2) 1
= π – sin-
2 2 (^) x 1 y y 1 x if x> 0 & y < 0 & x (^2) +y (^2) > 1
= - π – sin-
2 2 (^) x 1 y y 1 x if y > 0 &x < 0 & x (^2) +y (^2) > 1
- cos-1x +cos-1y =cos-1[xy 1 x 1 y ] 2 2 (^) if -1 x, y 1 , x + y 0
= 2π - cos -1[xy 1 x 1 y ] 2 2 (^) if -1 x, ,y 1 , x + y 0
- cos-1x _^ cos-1y =cos-1[xy 1 x 1 y ] 2 2 (^) if -1 x, y 1 , x y
= - cos-1[xy 1 x 1 y ] 2 2 (^) if -1 y 0 , 0 x 1 , x y
- 2 sin-1x = sin-
2 (^2) x 1 x if - 1/√2 x 1/√
= π - sin-
2 (^2) x 1 x if 1/ √2 x 1
= - π - sin-
2 (^2) x 1 x if -1 x - 1/√
- 2cos-1x = cos-1[2x^2 -1] if 0 x 1 = 2π - cos-1[2x^2 – 1] if -1 x 0
- 3sin-1x = sin-1[3x-4x^3 ] , if -1/2 x 1/
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
= π - sin-1^ [3x – 4x^3 ] if 1/2 < x 1 = - π - sin-1^ [3x – 4x^3 ] if -1 x < - 1/
- 3cos-1x = cos-1[4x^3 – 3x] , if 1/2 x 1
= 2 π - cos
- [4x 3 - 3x] , if - 1/2 x (^) 1/2 = 2 π + cos - [4x 3 - 3x] if -1 x (^) - 1/
Cosine rule
(i) Cos A = 2 bc
b c a
2 2 2
(ii) Cos B = 2 ac
a c b 2 2 2
(iii) Cos C = 2 ab
b a c 2 2 2
- Projection Formulae
(iv) a = b cosC + c cosB
(v) b = c cosA + a cosC
(vi) c = a cosB + b cosA
- Napier’s Analogy
(a) 2
C
Cot a b
a b
2
A B
Tan
(b) 2
A
Cot b c
b c
2
B C
Tan
(c) 2
B
Cot c a
c a
2
C A
Tan
- Semi Sum Formulae
(a) bc
(s b)(s c)
2
A
Sin
(b) bc
s(s a)
2
A
Cos
(c) s(s a)
(s b)(s c)
2
A
Tan
- Area of Triangle ABC is caSinB 2
bcSinA 2
abSinC 2
- Let R, r, r 1 , r 2 , r 3 be radii of circumcircle, incircle, excircles of ABC then
(a) 2 SinC
c
2 SinB
b
2 SinA
a R
(b)
abc R s
r
(c) s a
r 1
s b
r 2
s c
r 3
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
BASIC
- Which of the following is true
(a) tan 2 > tan 2 (b) sin 2 > sin 2
(c) cos 2 > cos 2 (d) N/T
- Which of the following is true
(a) sin 1 <sin 2 < sin 3
(b) sin 3 < sin 2 < sin 1
(c) sin 2 <sin 1 < sin 3
(d) sin 3 < sin 1 < sin 2
- Number of solutions of sin^8 x = 1+ tan^4 x
(a) 0 (b) l
(c) 4 (d) N/T
- Smallest +ve x satisfying
log (^) cos xsinxlogsinxcosx 2
(a) π/2 (b) π /
(c) π /4 (d) π /
- If tan-1^ ( sin^2 x ) > 1 then x
(a) ( 1/√2 , 1) (b) ( 0, 1/√2 )
(c) ( -1 , -1/√2 ) U ( 1/√2 , 1) (d) N/T
- Number of solutions of | √3 cos x – sinx | 2
In [ 0 , 4π ]
(a) 5 (b) 4
(c) ∞ (d) N/T
- General solution of sin^40 x – cos^40 x = 1 is
(a) nπ + π /3 (b) nπ + π /
(c) 2nπ + π /3 (d) 2nπ + π /
- Number of solutions of [ sinx ] = cosx , x [ 0 , 100 π ] is
(a) 1 (b) 2
(c) 3 (d) 0
- If cosx = cosy & sinx = - siny then sin 2006x+ sin2006y is
(a) 0 (b) 2006
(c) information insufficient
(d) N/T
- If x = 2 2 2 2 cos then x is
(a) 2 cos 10^0 (b) 2 cos 20^0
(c) 2 cos 40^0 (d) 2 cos 80^0
- If 4n = , then the value of tan tan 2 tan
3 tan 4 …… tan (2n – 2) tan(2n – 1) is equal to
(a) 0 (b) 1
(c) – 1 (d) N/T
- If sec and cosec are the roots of x^2 - px + q = 0, then
(a) p^2 = q(q – 2) (b) p^2 = q(q + 2)
(c) p^2 + q^2 = 2q (d) N/T
cos 11
cos 11
cos 11
cos 11
cos
(a) 0 (b) – 1/
ASSIGNMENT
TRIGONOMETRY
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
(c) 1/3 (d) N/T
cos 6 6 cos 4 15 cos 2 10
cos 7 7 cos 5 21 cos 3 35 cos
is equal to
(a) 2 cos (^) (b) cos
(c) 2
cos (^) (d) N/T
1 cos 8
1 cos 8
1 cos 8
1 cos is
(a) 2
(b) cos 8
(c) 8
(d) 2 2
- sin^2 8
sin 8
sin 8
sin 8
(^) is
(a) 2 (b) 0
(c) 1 (d) N/T
- If sin A + sin B = 0= cos A + cos B , then
cos 2A + cos 2B is equal to
(a) -2sin ( A + B) (b) 2sin ( A + B)
(c) -2cos ( A + B) (d) 2cos ( A + B).
- In a triangle ^ 2
C
sin 2
B
sin 2
A
cos 2 2 2
(a) 2
C
cos 2
B
cos 2
A
2 sin (b) 2
C
sin 2
B
sin 2
A
2 cos
(c) 2
C
sin 2
B
sin 2
A
2 sin (d) N/T
- If tan A is integral solution of 4x^2 – 16x + 15 <
0 and cos B is slope of bisector of first quardrant then sin (A + B) sin (A - B) =
(a) 4/5 (b) – 4/
(c) 1/5 (d) N/T
..........nterms n
sin n
sin n
sin
(a) n/2 (b) 2n/
(c) 1 (d) N/T
- If
2 sin
then 2 cot 4
is
(a) 1 + cot (b) – (1 + cot )
(c) 1 - cot (d) - 1 + cot .
- If y =
sec tan
sec tan 2
2 , then
(a) y 3 3
(b)
y
(c) – 3 < y <– 3
(d) N/T
- If an angle is divided into two parts A and B such that A – B = x and tan A : tan B = k : 1, then sin x is
(a)
sin k 1
k 1 (b)
sin k 1
k
(c)
sin k 1
k 1 (d) N/T
- The equation a sin x + b cos x = c, where | c | >
2 2 a b has
(a) a unique solution
(b) Infinite number of solutions.
(c) No solution
(d) None of these.
- Maximum of 3+ , 4
2 cos x 4
sin x
is
(a) 5 (b) 6
(c) 7 (d) N/T
- If A = cos^2 + sin^4 then for all , A
(a) [1, 2] (b) [13/16, 1]
(c) [3/4, 13/16] (d) N/T
- Equation sin^6 x + cos^6 x = , has a solution if
(a)
(b)
(c) 1 , 1 (d)
- Minimum f ( x ) =( 3sinx - 4 cos x – 10 ) (3sinx + 4 cos x – 10 ) is
(a) 84 (b) 45
(c) 4 9 (d) N/T
- Maxmum of 5 cos (^) +3 cos
- 3 is
(a) 5 (b) 10
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
- If the complex numbers sin x + i cos 2x and
cos x – i sin 2x are conjugate to each other, then x is equal to
(a) n (b)
n
(c) 0 (d) N/T
- The number of solutions of 2 cos^2 sin x 2
x (^2)
x^2 + 2
, 0 x x
2
is
(a) 0 (b) 1
(c) infinite (d) N/T.
- Number of solution for |cot x| = cot x + cosec
x in [ 0 ,2π ] is
(a) 0 (b) 1
(c) 2 (d) 3
- Number of solution for |sinx| = | cos3x | in
[ - 2π , 2π ] is
(a) 30 (b) 24
(c) 28 (d) 32
- General solution for |sin x| = cos x is
(a) 3
2 n
(b) 4
n
(c) 4
2 n
(d) N/T
- General solution for 2cot^2 x + 2√3 cotx + 4
cosec x + 8 =0 is
(a) 2 n / 6 (b) n / 6
(c) n / 6 (d) 2 n / 6
- If the solutions for of cos p + cos q = 0,
p > 0, q > 0 are in A.P., then the numerically smallest common difference of A.P. is
(a) p q
(b) p q
(c)
2 pq
(d) p q
- If tan sec 3 ; 0 ,thenis
(a) /3 (b) 2 /
(c) /6 (d) 5 /8.
- ,(acute) satisfy sin =1/2 cos =1/3then
+ :
(a)
2
, 3
(b)
(c)
(d)
- The real roots of equation cos^7 x + sin^4 x = 1in
the interval ( - , ) are :
(a) - , 0 2
(b) - , 0 2
(c) , 0 2
(d) 0, 4
INVERSE
- cos-1cos
(a) 3
(b) 3
(c) 3
(d) N/T
- tan-11 + tan-12 + tan-13 =?
(a) 0 (b) 2
(c) (d) N/T
- sin-1x + sin-1y = 2 3 then cos-1x + cos-1y =?
(a) 6 (b) 3
(c) 2 3 (d)
- Sin-1^ (sin 4) is
(a) 4 (b) π -
(c) 4 -π (d) N/T
- The value of
3
tan 3
cot cosec 1 1 is
(a) 3/17 (b) 4/
(c) 5/17 (d) 6/
- If x = sin (2 tan-1^ 2) and
3
tan 2
sin
1 y , th
(a) x > y and y^2 = 1 – x(b) x < y
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
(c) x > y and y^2 = x (d) y^2 = 1 + x
- Solution set of equation sin-1x = 2 tan-1x, is
(a) {1, 2} (b) {-1, 2}
(c) {-1, 1, 0} (d) {1, 1/2, 0}
- If cos-1x + cos-1y + cos-1z = 3 then xy + yz +
zx =?
(a) – 3 (b) 0
(c) 3 (d) – 1
- If sin-1x + sin-1y = then x^100 + y^1005 =?
(a) 0 (b) 2
(c) 1 (d) – 1
- If sin-1x + sin-1y + sin-1z = 2
then x^2 + y^2 + z^2 +
2xyz =?
(a) 0 (b) 1
(c) – 1 (d) N/T
- If x^2 + y^2 + z^2 = r^2 , then
yr
xz tan xr
yz tan zr
xy tan
1 1 1 is equal to
(a) (b) /
(c) 0 (d) N/T
- tan^2 (sec-12) + cot^2 (cosec-14) =?
(a) 13 (b) 18
(c) 17 (d) N/T
- sin (cot-1(tancos-1x)) =?
(a) x (b) 2 1 x
(c) 1/x (d) N/T
- sin-
1
4
x
2
x cos x 4
x
2
x x
4 6 1 2
2 3 (^) =
2 then x =?
(a) 1/2 (b) – 1
(c) – 1/2 (d) 1
- If cos-1 / 6 then 3
y cos 2
x (^1)
9
y
2 3
xy
4
x cos
2 2 1 is
(a) 3/4 (b) 1/
(c) 1/4 (d) N/T
- tan x(x 1 ) sin x x 1 / 2 1 1 2 , No. of
Solns.
(a) 0 (b) 1
(c) 2 (d)
- tan
a
b cos 2
1
- tan
a
b cos 2
1
(a) 2a/b (b) a/b
(c) 2b/a (d) b/a
- If cos x 0 then x?
20
i 1
i i
20
i 1
1 ^
(a) 0 (b) 10
(c) 20 (d) 5
- If 0< a < b < c then cot -1^ ( ) a b
1 ab
- cot -1^ (
b c
1 bc cot -1^ ( ) c a
1 ac
(a) 0 (b) π
(c) 2π (d) N/T
- If a> b> c> 0 then cot -1^ ( ) a b
1 ab
cot -1^ (
b c
1 bc cot -1^ ( ) c a
1 ac
(a) 0 (b) π
(c) 2π (d) N/T
- If p > q > 0 & pr < -1 < qr then tan- 1 pq
p q
- tan- 1 rq
q r
(^) + tan- 1 pr
r p
(^) is
(a) 0 (b) π
(c) cant say (d) - π
- If 0 < x < 1 &1 + sin-1^ x + (sin-1^ x )^2 + (sin-1^ x )^3
- --------∞ = 2 then sin-1x is
(a) π/6 (b) π/
(c) π /12 (d) π / 4
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
(a) r + R (b) r – R
(c) 2(r + R) (d) N/T
- a cos A + b cos B + c cos C is
(a) 8∆^2 / abc (b) 4∆^2 / abc
(c) 3∆^2 / abc (d) N/T.
- If ∆ABC is a right triangle, the value of
A-B C
2 acsin can be equal to
(a) 4r^2 (b) 8R^2
(c) 2 R
(d) R
r
- If cos(A C)
cos(A C) cos 2 B
then tan A,tan B,tan C
are in
(a) A.P. (b) G.P.
(c) H.P. (d) N/T
- If altitudes of a triangle are in AP then sides are in
(a) AP (b) HP
(c) GP (d) AGP
- If p 1 , p 2 , p 3 are the altitudes of a triangle ABC from the vertices A, B, C and the area of the
triangle, then p 2 3
2 2
2 1 p^ p
is equal to
(a)
a bc (b) 2
2 2 2
a b c
(c) 2
2 2 2 a b c
(d) N/T
- If cos A + cos B + 2 cos c = 2 then sides are in
(a) AP (b) GP
(c) HP (d) N/T
- In a ABC, a = 1 & perimeter is six times the
AM of sines of angles then A=?
(a) / 3 (b) /
(c) /4 (d) /
- In ABC two larger sides are 10 & 9. If angles
are in A.P then 3rd^ side is
(a) 3 3 (b) 5^6
(c) 5 (d) N/T
- If the radius of circumcircle of an isosceles triangle PQR is equal to PQ ( = PR ), then the angle P is
(a) π/6 (b) π/
(c) π/2 (d) 2π/3.
- In a triangle ABC with sides a, b, c, r 1 > r 2 > r 3 (which are the ex-radii ), then
(a) a > b > c (b) a < b < c
(c) a> b and b< c (d) a< b and b > c.
- r 1 =2 r 2 = 3 r 3 then
(a) a/b = 4/5 (b) a/b = 5/
(c) a+ b = 2c (d) 2a = b+c
- If then r
r
r
r
3
2
1
(a) A 90 (b) B 90
(c) C 90 (d) N/T
- In any harmonic mean of ex-radii is
(a) 3r (b) 2R
(c) R + r (d) N/T
- r 1 r 2 + r 2 r 3 + r 3 r 1
(a) S^2 (b) 2S^2
(c) 3S^2 (d) N/T
- If b + c = 3a, then the value of cot 2
B
cot 2
C
is
(a) 1 (b) 2
(c) 3 (d) 2.
- In a triangle ABC (Sin A + Sin B + Sin C) (Sin A
+ Sin B - Sin C) = 3 Sin A SinB then angle C (a) /6 (b) /
(c) /3 (d) N/T
- The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is :
(a)
2 n
cot 4
a (b) a cot
n
(c)
2 n
cot 2
a (d) a cot
2 n
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
- If two towers of hts b 1 and b 2 subtend 60 and
30 at mid-point of line joining their feet, then b 1 : b 2 =
(a) 1 : 2 (b) 1 : 3
(c) 2 : 1 (d) 3 : 1
- A man from the top of a 100 m high tower
sees a car moving towards the tower at an angle of depression of 30. After some time,
the angle of depression becomes 60. The distance (in metres) travelled by the car during this time is
(a) 100 3 (b) 3
(c) 3
(d) 200 3
- A person standing on bank of a river observes
that angle of elevation of top of a tree on opposite bank is 60 and when he retires 40 meters away from the tree, elevation is 30. The breadth of the river is :
(a) 20 m (b) 30 m
(c) 40 m (d) 60 m.
- In ABC, sides a,b,c are in A.P., a being
smallest, then cosA is
(a) 2 c
3 c 4 b (b) 2 b
3 c 4 b
(c) 2 c
4 c 3 b (d) N/T
- In triangle ABC, 3sinA + 4cosB = 6 and 4sinB
- 3cosA = 1. Then the measure of the angle C
(a) 30° (b) 150°
(c) 30° or 150° (d) N/T,
sin
sin sin 3
cos
cos cos 3
3 3
(a) 0 (b) 3
(c) 1 (d) 5
- If cos x + cos y + cos α =0 & If sinx + siny +
sin α =0 then cot
x y is
(a) sinα (b) cos α
(c) cot α (d) 2 sinα
- Let a = (tan π/8) tan π/8^ , b = (tan π/8) cotπ/8^ , c = (cot π/8 ) tan π/ 8^ & d = (cot π/8 ) cot π/ 8
Then which one of the following statements is true about relative sizes of a,b,c,d?
(a) d > c > b> a (b) c > d > b > a,
(c) d > c > a > b (d) c > a > b > d
- Number of solutions of pair of 2 sin^2 θ - cos2θ = 0 , 2 cos^2 θ – 3sinθ = 0 in [ 0, 2π ] is
(a) 1 (b) 0
(c) 2 (d) 4
- cot -1^ (1^2 + 3/4 ) + cot -1^ ( 2^2 + 3/4 ) + cot -1^ ( 32 + 3/4 ) + ……∞
(a) π /4 (b) tan-1^2
(c) tan-1^3 (d) N/T
- If sin^2 x – 2 sinx – 1=0 has exactly 6 roots in [ 0 , n π ] then minimum of n is
(a) 2 (b) 4
(c) 6 (d) 3
- A triangular field has fencing of length x each on two of its sides while third is on river bank then maximum area is
(a) x^2 (b) x^2 /
(c) x^2 √2 (d) x^2 / √
- If sin A – sin B = a & cos A + cos B = b then
(a) a2 +^ b 2 4 (b) a2 +^ b 2 4
(c) a2 +^ b 2 3 (d) a2 +^ b 2 2
- If 0 a 3, 0 b 3, & x 2 + 4 + 3 cos ( ax + b) = 2x has atleast one solution then a + b is
(a) 0 (b) π / 2
(c) π (d) N / T
- For ∆ ABC sin 2 A + sin 2 B + sin 2 C - 2 cos A cos B cos C is
(a) 0 (b) 2
(c) -2 (d) -
- Number of points inside or on x^2 + y^2 = satisfying tan^4 x + cot 4 x + 1 = 3 sin^2 y is
(a) 0 (b) 2
(c) 4 (d) ∞
- There exists a ABC satisfying the conditions
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
ANSWER (TRIGONOMETRY)
b d a c c b b d a c b b b b c
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
a c c a d b a c c b b b a b c
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
b a c b a c a d c c c b c c a
d a c b c d b c b b a c b b d
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
a c c b b b b a d c c a c c B
b a b a b c d a a a c a b b a
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
d a b c a b b b b a b b d a b
c a a d c c d b c c a b b c c
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
b c b b c b c a b a c c c c c