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7/27/2019 Trigonometrygdfgdfgdg

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TRIGONOMETRY[ BASIC, EQUATIONS, INVERSE, SOLUTION OF ,HT & DISTANCES]By:- Nishant Gupta

For any help contact:

9953168795, 9268789880


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Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85

Contact: 9953168795, 9268789880

BASICS

Some Formulae

1.4

A3SinA

3SinA

3SinASin

2.

4

A3Cos

A3CosA3CosACos

3. A3TanA3

TanA3

TanATan

4. tan+ tan

3tan3

3tan

3

5. 2/3A3

SinA3

SinASin 222

6. BSinASinBASin)BA(Sin 22 7. BSinACosBACos)BA(Cos 22 8.

SinA2

A2Sintermsn...........A4ACos2CosACos

n

n

9.n2

1

1n2

ncos........

1n2

3cos

1n2

2cos

1n2cos

10. In a triangle ABCTanCTanBTanAnCTanATanBTa &

2

CCot

2

BCot

2

ACot

2

CCot

2

BCot

2

ACot

11. Sin + Sin( + ) + Sin( + 2) + + Sin( + 1n ) =2

Sin

2

nSin}

2

1n{Sin

TRIGONOMETRY


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12. Cos + Cos (+ ) +Cos ( + 2) + + Cos ( + 1n ) =2

Sin

2

nSin}

2

1n{Cos

13. (a) Maximum & Minimum of a cosA + b sinA are 22 ba and 22 ba resp(b) The minimum value of iseccosseca

2222

2

,0whereba2

(c) The minimum value of a sec + b cosec is

2,0where.ba

2/33/23/2 .

(d) The minimum value of a sin + b cosec is ,0where.ab2 .

(e) The minimum value of a sec + b cos is

2,

2where.ab2

(f) The minimum value of a tan + b cot is

2

3

,or2,0where.ab2 .

Trigonometrical Equations

(a) SinA = 0 A = n

(b) CosA = 0 A = (2n + 1)/2

(c) TanA = 0 A = n

(d) Sin A = SinB A = n + (-1)nB

(e) Tan A = TanB A = n + B

(f) Cos A = Cos B A = 2n B

(g) Sin2A = Sin2B ; Cos2A = Cos2B ; Tan2A = Tan2B all A = n B

Inverse Trigonometry

Principal values sin-1x

2,

2

cos-1x [0, ] sec-1x [0, ] except /2

tan-1x (-/2, /2)

cosec-1x

2,

2except 0 cot-1x (0, )

If angle is principal one then sin-1sinx = x cos-1cosx = x etc

14. sin-1x + cos-1x =/2, cosec-1x+sec-1x= /2 , tan-1x + cot-1x= /2

15. sin-1x

1= cosec-1x, cot-1x =

0x,x

1tan

0x,x

1tan

1-

1-

cos-1x = sec-1 1/x etc

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16. sin-1(x) = sin-1x cosec-1 (x)= cosec-1x sec-1 (x) = sec-1xtan-1(x) = tan-1x cot-1 (x) = cot-1x cos-1(x) = cos-1x

17. tan-1x + tan-1y = tan-1xy1

yx

, x y < 1 = + tan-1xy1

yx

, x y > 1, x > 0

= - + tan-1xy1yx

, x y > 1,x < 0

= / 2 , x y = 1 , x > 0 = - / 2 , x y = 1 , x < 0

18. tan-1x - tan-1y = tan-1xy1

yx

, x y > -1 = + tan-1

xy1

yx

, x y < -1 , x > 0

= - + tan-1xy1

yx

, x y < -1 , x < 0

= /2 , x y = -1 , x > 0 = - / 2 , x y = -1 , x < 0

19. sin-1x + sin-1y = sin-1

22 x1yy1x 0 or (x y > 0 & x2 +y2 1)

= sin-1

22 x1yy1x if x & y > 0 & x2 +y2 > 1

= - sin-1

22 x1yy1x if x & y < 0 & x2 +y2 >1

20. sin-1x- sin-1y =sin-1 22 x1yy1x if xy 0 or (x y) > 0 & x2 +y2 1= sin-1

22 x1yy1x if x> 0 & y < 0 & x2 +y2> 1

= - sin-1

22 x1yy1x if y > 0 &x < 0 & x2 +y2> 1

21. cos-1x +cos-1y =cos-1 ]y1x1xy[ 22 if -1 x, y 1 , x + y 0= 2 - cos -1 ]y1x1xy[ 22 if -1 x, ,y 1 , x + y 0

22. cos-1x _ cos-1y =cos-1 ]y1x1xy[ 22 if -1 x, y 1 , x y= - cos-1 ]y1x1xy[ 22 if -1 y 0 , 0 x 1 , x y

23. 2 sin-1x = sin-1 2x1x2 if -1/2 x 1/2= - sin-1

2x1x2 if 1/ 2 x 1

= - - sin-1

2x1x2 if -1 x - 1/2

24. 2cos-1x = cos-1[2x2-1] if 0 x 1 = 2 - cos-1[2x2 1] if -1 x 025. 3sin-1x = sin-1[3x-4x3] , if -1/2 x 1/2

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= - sin-1 [3x 4x3] if 1/2 < x 1 = - - sin-1 [3x 4x3] if -1 x < - 1/2

26. 3cos-1x = cos-1[4x33x] , if 1/2 x 1= 2 - cos

-1[4x

33x] , if - 1/2 x 1/2 = 2 + cos

-1[4x

33x] if -1 x - 1/2

Cosine rule

(i) Cos A = bc2acb222

(ii) Cos B =ac2

bca 222

(iii) Cos C =ab2

cab 222

27. Projection Formulae(iv) a = b cosC + c cosB(v) b = c cosA + a cosC(vi) c = a cosB + b cosA

28. Napiers Analogy(a)

2

CCot

ba

ba

2

BATan

(b)2

ACot

cb

cb

2

CBTan

(c)2

BCot

ac

ac

2

ACTan

29. Semi Sum Formulae(a) bc

)cs)(bs(

2

ASin

(b)bc

)as(s

2

ACos

(c))as(s

)cs)(bs(

2

ATan

30.Area of Triangle ABC is caSinB2

1bcSinA

2

1abSinC

2

1

31. Let R, r, r1, r2, r3 be radii of circumcircle, incircle, excircles of ABC then(a)

SinC2c

SinB2b

SinA2aR

(b)

4

abcR

sr

(c)as

r1

bs

r2

;

csr3


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(d)2

Atan)as(r

2

Btan)bs(r ;

2

Ctan)cs(r

(e)2

Atansr1

2

Btansr2 ;

2

Ctansr3

(f) R4rrrr 321

(g)r

1

r

1

r

1

r

1

321

(h) 2133221 srrrrrr

(i)R

r1CcosBcosAcos

(j)R

sCsinBsinAsin

32. Regular polygon & radii of the inscribed & circumscribing circles of a regular polygon :Regular Polygon - It is a polygon whose sides are equal and also its angles are equal.

(i) R (radius of the circum circle of a regular polygon of n sides) =n

eccos2

a

(ii) R ( radius of the in circle of a regular polygon of n sides) = ,n

cot2

a where a

is the length of each side of the polygon.

(iii) Area of a regular polygon of n sides =n

2sinR

2

n

ncotna

4

1 22

=n

tannr2


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Contact: 9953168795, 9268789880

BASIC

1. Which of the following is true(a) tan 2 > tan 2 (b) sin 2 > sin 2

(c) cos 2 > cos 2 (d) N/T

2. Which of the following is true(a) sin 1

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(c) 1/3 (d) N/T

14.102cos154cos66cos

cos353cos215cos77cos

is equal to

(a) 2 cos (b) cos

(c) 2

1

cos (d) N/T

15.

8

7cos1

8

5cos1

8

3cos1

8cos1 is

(a)2

1(b) cos

8

(c)8

1(d)

22

21

16. sin28

7sin

8

5sin

8

3sin

8

222

is

(a) 2 (b) 0

(c) 1 (d) N/T

17. If sin A + sin B = 0= cos A + cos B , thencos 2A + cos 2B is equal to

(a) -2sin ( A + B) (b) 2sin ( A + B)

(c) -2cos ( A + B) (d) 2cos ( A + B).

18. In a triangle 2

Csin

2

Bsin

2

Acos 222

(a) 2

C

cos2

B

cos2

A

sin2 (b) 2

C

sin2

B

sin2

A

cos2

(c)2

Csin

2

Bsin

2

Asin2 (d) N/T

19. If tan A is integral solution of 4x2 16x + 15 0, q > 0 are in A.P., then the numerically

smallest common difference of A.P. is

(a)qp

(b)

qp

2

(c) qp2

(d) qp

1

53. If tan then,0;3sec is(a) /3 (b) 2 /3

(c) /6 (d) 5 /8.

54. ,(acute) satisfy sin =1/2 cos =1/3then + :

(a)

2

,3

(b)

3

2,

2

(c) 65,

32 (d) ,6

5

55. The real roots of equation cos7 x + sin4 x = 1inthe interval ( - , ) are :

(a) - 0,2

(b) - 0,

2

,2

(c) 0,2

(d) 0,

4

,2

INVERSE

56. cos-1cos

3

4= ?

(a)3

(b)

3

(c)3

4(d) N/T

57. tan-11 + tan-12 + tan-13 = ?(a) 0 (b) 2

(c) (d) N/T58. sin-1x + sin-1y = 32 then cos-1x + cos-1y = ?

(a) 6 (b) 3

(c) 32 (d)

59. Sin-1 (sin 4) is(a) 4 (b) -4

(c) 4 - (d) N/T

60. The value of

3

2tan

3

5eccoscot 11 is

(a) 3/17 (b) 4/17

(c) 5/17 (d) 6/17

61. If x = sin (2 tan-1 2) and

3

4tan

2

1sin 1y , th

(a) x > y and y2 = 1 x(b) x < y

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(c) x > y and y2 = x (d) y2 = 1 + x

62. 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}

63. If cos-1x + cos-1y + cos-1z = 3 then xy + yz +zx = ?

(a) 3 (b) 0

(c) 3 (d) 1

64. If sin-1x + sin-1y = then x100 + y1005 = ?(a) 0 (b) 2

(c) 1 (d) 1

65. If sin-1x + sin-1y + sin-1z =2

then x2 + y2 + z2 +

2xyz = ?

(a) 0 (b) 1

(c) 1 (d) N/T

66. If x2 + y2 + z2 = r2 , thenyr

xztan

xr

yztan

zr

xytan 111

is equal to

(a) (b) /2

(c) 0 (d) N/T

67. tan2 (sec-12) + cot2(cosec-14) = ?(a) 13 (b) 18(c) 17 (d) N/T

68. sin (cot-1(tancos-1x)) = ?(a) x (b) 2x1

(c) 1/x (d) N/T

69. sin-1

4

x

2

xxcos

4

x

2

xx

6421

32

=

2 then x = ?

(a) 1/2 (b) 1(c) 1/2 (d) 1

70. If cos-1 then6/3

ycos

2

x 1

9

y

32

xy

4

xcos

221 is

(a) 3/4 (b) 1/2

(c) 1/4 (d) N/T

71. 2/1xxsin)1x(xtan 211 , No. ofSol

n

s.(a) 0 (b) 1

(c) 2 (d)

72. tan

a

bcos

2

1

4

1 + tan

a

bcos

2

1

4

1

(a) 2a/b (b) a/b

(c) 2b/a (d) b/a

73. If ?xthen0xcos 201i

ii

20

1i

1

(a) 0 (b) 10(c) 20 (d) 5

74. If 0< a b> c> 0 then cot -1 ( )ba

ab1

+

cot-1 ( )

cb

bc1 cot-1 ( )ac

ac1

(a) 0 (b)

(c) 2 (d) N/T

76. If p > q > 0 & pr < -1 < qr then tan-1pq1

qp

+ tan-1rq1

rq

+ tan-1

pr1

pr

is

(a) 0 (b)

(c) cant say (d) - 77. If 0 < x < 1 &1 + sin-1 x + (sin-1 x )2 + (sin-1 x )3

+ -------- = 2 then sin-1x is

(a) /6 (b) /3

(c) /12 (d) / 4

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78. Solution of the equation

2

21

2

1

x1

x1cos4

x1

x2sin3

3x1

x2tan2

2

1

for x is

(a) x = 3 (b) x = 1/ 3

(c) x = 1 (d) x = 0.

79.

1r

214/3rcot is

(a) tan -1 2 (b) cos -1 2

(c) 2 tan-1 1 (d) N/T

80. Range of sin-1 x + tan-1 x + cos-1 x is(a) [ 0 , ] (b) [ /4 , 3/4 ]

(c) [ - /4 , 3/4 ] (d) [- /4 , /2 ]

SOLUTION OF ( NOT IN SYLLABUS)

81. In a triangle ABC, if a,b,c are in A.P., thentan

2

A, tan

2

B, tan

2

C, are in

(a) A.P. (b) G.P.

(c) H.P. (d) N/T

82. Two straight roads intersect an angle of 60.A bus on one road is 2 km away from the

intersection and a car on the other is 3 km

away from the intersection ; then the direct

distance between the two vehicles is

(a) 1 km (b) 2 km

(c) 4 km (d) 7 km

83. In a ABC, C =2

, then 2 ( r + R ) =

(a) a + b (b) b + c

(c) c + a (d) a + b + c.

84. The sides of a triangle are 3x + 4 y, 4x +3yand 5x + 5y where x,y > 0, then the triangle is

(a) obtuse angled (b) right angled

(c) equilateral (d) none of these.

85. If 0 < x r2 > r3(which are the ex-radii ), then

(a) a > b > c (b) a b and b< c (d) a c.88. In ABC Asin/)1AsinA(sin2 is greater

than

(a) 3 (b) 9

(c) 27 (d) N/T

89. If a = 2b & |A-B| = cthen3

= ?

(a) /4 (b) /3

(c) /6 (d) N/T

90. In ABC sin A + sin B + sin C = ?(a) S/R (b)R/S

(c) r/R (d) N/T

91. If 3a = b + c then tan2

Ctan

2

Bis

(a) tan( A/2) (b)1

(c) 2 (d) N/T

92. If c cos22

b3

2

Ccosa

2

A 2 then a,b,c are in

(a) AP (b) GP

(c) HP (d) N/T

93. If)CBsin(

)BAsin(

Csin

Asin

then

(a) a, b, c AP (b) a2, b2, c2AP

(c) a, b, c HP (d) N/T

94. a cot A + b cot B + c cot C = ?

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(a) r + R (b) r R

(c) 2(r + R) (d) N/T

95. a cos A + b cos B + c cos C is(a) 82 / abc (b) 42 / abc

(c) 32 / abc (d) N/T.

96. If ABC is a right triangle, the value of

2

CB-Asinac2 can be equal to

(a) 4r2 (b) 8R2

(c)R2

1(d)

R

1

r

1

97. If)CAcos(

)CAcos(B2cos

then tan A,tan B,tan C

are in

(a) A.P. (b) G.P.(c) H.P. (d) N/T

98. If altitudes of a triangle are in AP then sidesare in

(a) AP (b) HP

(c) GP (d) AGP

99. If p1, p2, p3 are the altitudes of a triangle ABCfrom the vertices A, B, C and the area of thetriangle, then p 23

22

21 pp

is equal to

(a)

cba

(b) 2

222

4

cba

(c)2

222 cba

(d) N/T

100. If cos A + cos B + 2 cos c = 2 then sides are in(a) AP (b) GP

(c) HP (d) N/T

101. In a ABC, a = 1 & perimeter is six times theAM of sines of angles then A = ?

(a) 3/ (b) /6

(c) /4 (d) /2102. 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

103. If the radius of circumcircle of an isoscelestriangle PQR is equal to PQ ( = PR ), then the

angle P is

(a) /6 (b) /3

(c) /2 (d) 2/3.

104. In a triangle ABC with sides a, b, c, r1 > r2 > r3(which are the ex-radii ), then(a) a > b > c (b) a b and b< c (d) a c.

105. r1 =2 r2 = 3 r3 then(a) a/b = 4/5 (b) a/b = 5/4

(c) a+ b = 2c (d) 2a = b+c

106. If thenr

r

r

r

3

2

1

(a) 90A (b) 90B

(c) 90C (d) N/T

107. In any harmonic mean of ex-radii is(a) 3r (b) 2R

(c) R + r (d) N/T

108. r1r2 + r2r3 + r3r1(a) S2 (b) 2S2

(c) 3S2 (d) N/T

109. If b + c = 3a, then the value of cot2

Bcot

2

Cis

(a) 1 (b) 2

(c) 3 (d) 2 .

110. 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) /4

(c) /3 (d) N/T

111. The sum of the radii of inscribed andcircumscribed circles for an n sided regular

polygon of side a, is :

(a)

n2cot

4

a(b) a cot

n

(c)

n2cot

2

a(d) a cot

n2


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112. If two towers of hts b1 and b2 subtend 60 and30 at mid-point of line joining their feet, thenb1 : b2 =

(a) 1 : 2 (b) 1 : 3

(c) 2 : 1 (d) 3 : 1

113. A man from the top of a 100 m high towersees a car moving towards the tower at anangle of depression of 30. After some time,the angle of depression becomes 60. Thedistance (in metres) travelled by the car

during this time is

(a) 100 3 (b)3

3200

(c)3

3100(d) 200 3

114. A person standing on bank of a river observesthat angle of elevation of top of a tree on

opposite bank is 60 and when he retires 40meters 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.

115. In ABC, sides a,b,c are in A.P., a beingsmallest, then cosA is

(a)c2

b4c3 (b)

b2

b4c3

(c)c2

b3c4 (d) N/T

116. 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,

117.

sin

3sinsin

cos

3coscos 33

(a) 0 (b) 3

(c) 1 (d) 5118. If cos x + cos y + cos =0 & If sinx + siny +

sin =0 then cot

2

yxis

(a) sin (b) cos

(c) cot (d) 2 sin

119. Let a = (tan /8) tan /8 , b = (tan /8) cot/8 ,c = (cot /8 )tan / 8 & d = (cot /8 ) cot / 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 > d120. Number of solutions of pair of 2 sin2 -

cos2 = 0 , 2 cos2 3sin = 0 in [ 0, 2 ] is

(a) 1 (b) 0

(c) 2 (d) 4

121. cot-1 (12 + 3/4 ) + cot-1 ( 22 + 3/4 ) + cot -1 (32 + 3/4 ) +

(a) /4 (b) tan-1 2

(c) tan-1 3 (d) N/T

122. If sin2 x 2 sinx 1=0 has exactly 6 roots in [0 , n ] then minimum of n is

(a) 2 (b) 4

(c) 6 (d) 3

123. A triangular field has fencing of length x eachon two of its sides while third is on river bank

then maximum area is

(a) x2 (b) x2 /2

(c) x2 2 (d) x2 / 2

124. If sin A sin B = a & cos A + cos B = b then(a) a2 + b 2 4 (b) a2 + b 2 4

(c) a2 +

b2

3 (d) a2 +

b2

2125. 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) / 2

(c) (d) N / T

126. 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) -1

127. Number of points inside or on x2 + y2 =4satisfying tan4 x + cot4 x + 1 = 3 sin2 y is

(a) 0 (b) 2

(c) 4 (d)

128. There exists a ABC satisfying the conditions


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(a) abAaAb ,2

,sin

or2

,sin

Aaab .

(b)2

,sin

AaAb

(c)2

,sin

AaAb (d)2

,sin

AaAb

129. From the top of a h meter high cliff, angles ofdepression of the top and the bottom of a

tower are observed to be 30o and 60o

respectively. The height of the tower is

(a) h/3 (b) 2h/3

(c) 3h (d)3

h2

130. At a point 15 metres away from the base of a15 metres high house, the angle of elevation

of the to is

(a) 45 (b)

30

(c) 60 (d)

90

131. A tree is broken by wind and its upper parttouches the ground at point 10 metres from

the foot of the tree and makes an angle of 45

with the ground. The entire

(a) 15m (b) 20m

(c) m2110 (d) m

2

3110

132. If the sides of a triangle ABC are in A.P., and ais the smallest side, then cos A equal to:

(a)c

bc

2

43 (b)

b

bc

2

43

(c)c

bc

2

34 (d) N/T

133. If twice the square of the diameter of a circleis equal to half the sum of the squares of the

sides of inscribed triangle ABC, the sin2A +

sin2B + sin2C is equal to:

(a) 1 (b) 2

(c) 4 (d) 8

134. If 3sin2 A + 2sin2 B = 1 and 3sin 2A 2sin 2B =0, where A and B are acute angles, then

A + 2B is equal to:

(a)3

(b)

4

(c)2

(d) None of these

135. If in ABC the line joining the circumcentreand the incentre is parallel to BC then

(a) r = R sin A (b) R = r sin A

(c) r = R cos A (d) R = r cos A


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ANSWER (TRIGONOMETRY)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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 a46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

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

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

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

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