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M C A E N T R A N C E | I I T J E E | F O U N D A T I O N S
TRIGONOMETRIC RATIOS, IDENTITIES
AND MAXIMUM & MINIMUM VALUES OF
TRIGONOMETRICAL EXPRESSIONS
AQT1.1 INTRODUCTION
DEFINITION: An angle is the amount of rotation of a revolving line with respect
to a fixed line.
If the rotation is in clock-wise sense, the angle measured is negative and it
is positive if the rotation is in anti-clockwise sense.
There are three systems of measuring an angle viz.
1. Sexagesimal system or English system
2. Circular system
3. French system
First two of these three systems are commonly used. In sexadecimal
system, a right angle is divided into 90 equal parts called degrees. Further,
each degree is divided into sixty equal parts called minutes and each minute is
divided into sixty equal parts called seconds.
Thus
1 right angle = 90 degrees (90o)
1o=60 minutes (60’)
1o = 60 seconds (60”)
In circular system the unit of measurement is radian. One radian is the angle
made by an arc of length equal to radius of a given circle at its centre.
Relation between degree and radian. If D is the degree measure of an angle
and R is its measure in radians, then
RD 2
90
1radian =
180degrees = 57o 17’ 45” (approximately)
and 1 degree = 180
radian
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AQT1.2 SOME BASIC FORMULAE
1. sin2 A + cos2 = 1
2. 1+tan2 A = sec2 A or sec2 A – tan2 A = 1
Or sec A + tan A = 'tansec
1
AA where A ≠ n + ,
2
Z.
3. 1+cot2 A = cosec2 A or cose2 A-cot2 A = 1 or cosec A + cot A=
,cotcos
1
AecA where A ≠ n , n Z
AQT1.3 DOMAIN AND RANGE OF TRIGONOMETRICAL FUNCTIONS
Domain Range
sin A R [-1, 1]
cos A R [-1,1]
tan A R- {(2n+1) /2 | nZ} (-∞, ∞) = R
cosec A R – {(n |nZ} (-∞, -1][1, ∞
sec A R – { (2 n+1)/2 | nZ} (-∞, -1][1, ∞)
cot A R – {n |nZ} (-∞, ∞) = R.
Thus, |sin A| 1, |cos A| 1, sec A 1 or sec A -1and cosec A 1 or
cosec A -1.
AQT1.4 SUM AND DIFFERENCE FORMULAE
1. sin (A+B) = sin A cos B + cos A sin B
2. sin (A - B) = sin A cos B- cos A sin B
3. cos (A + B) = cos A cos B – sin A sin B
4. cos (A - B) = cos A cos B + sin A sin B
5. tan (A+B)=
BA
BA
tantan1
tantanwhre A≠n + ,
2
B≠ n +
2
6. tan (A-B)=
BA
BA
tantan1
tantanand A B ≠ m +
2
7. cot (A+B)=
BA
BA
cotcot
1cotcotwhere A ≠ n , B ≠ n
cot (A-B)=
AB
A
cotcot
1cotcotand A B ≠ n
8. sin (A+B) sin (A-B) = sin2 A – sin2 B = cos2 B-cos2A
9. cos (A+B) cos (A-B)= cos2 A - sin2 B = cos2 B-sin2 A
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10. sin 2 = 2 sin cos =
2tan1
tan2
11. cos 2 = cos2 - sin2 = 2 cos2 -1 = 1-2 sin2 = .tan1
tan12
2
12. 1+cos 2 = 2 cos2 , 1- cos 2 = 2 sin2 or 2
2cos1 = cos2 ,
2
2cos1
= sin2
13. tan 2 =
2tan1
tan2
, where ≠ (2n+1)
4
14. ,2
tansin
cos1
where ≠ 2n
15. ,2
cotsin
cos1
where ≠ (2 n + 1)
16. ,2
tancos1
cos1 2
where ≠ (2 n + 1)
17. ,2
cotcos1
cos1 2
where ≠2n
18. sin 3 = 3 sin - 4 sin3
19. cos3 = 4 cos3 - 3 cos
20. tan 3 =
2
3
tan31
tantan3
21. cos A cos 2A cos22 A … cos 2n-1 A =A
An
n
sin2
2sin
AQT1.5 SUM AND DIFFERENCE INTO PRODUCTS
1. Sin A + Sin B = 2 sin
2cos
2
BABA
2. Sin A + Sin B = 2 sin
2cos
2
BABA
3. cos A + cos B = 2 cos
2cos
2
BABA
4. cos A – cos B = - 2 sin
2sin
2
BABA
5. tan A+tan B = ,coscos
)sin(
BA
BAwhre A, B ≠ n +
2
6. tan A-tan B = ,coscos
)sin(
BA
BAnZ
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7. cos A+cot B = ,sinsin
)sin(
BA
BAwhere A, B ≠ n , n Z
8. cos A+cot B = ,sinsin
)sin(
BA
BAwhere A, B ≠ n , n Z
AQT1.6 PRODUCT INTO SUM OR DIFFERENCE
1. 2 sin A cos B = sin (A + B) + sin (A - B)
2. 2 cos A sin B = sin (A + B) - sin (A - B)
3. 2 cos A sin B = cos (A + B) + cos (A - B)
4. 2 sin A sin B = cos (A - B) – cos (A + B)
AQT1.7 T-RATIOS OF THE SUM OF THREE OR MORE ANGLES
1. sin (A+B+C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C –
sin A sin B sin C
or sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C –tan A tan B
tan C)
2. cos (A + B + C) = cos A cos B cos C – sin A sin B cos C – sin A cos B sin C
– cos A sin B sin C – sin A cos B sin C – cos A sin B sin C
3. tan (A + B + C) =A tan C tan-C tan B tan-B tan A tan1
C tan B tan A tan - C tan B tanA tan
4. sin (A1 + A2 + … + An)
= cos A1 cos A2 … cos An (S1 S3 S5 S7 S7 + …) = cos A1 coc A2 … cos An (S1-
S3 + S5 – S7 + …)
5. cos (A1 +A2 + …+ An) = cos A1 cos A2 … cos An (1- S2 + S4 – S6 + …)
6. tan (A1 + A2 + … An) = ,...1
...
642
7531
SSS
SSSSwhere
S1 = tan A1 +tan A2 + …+ tan An
= the sum of the tangents of the separate angles,
S2 = tan A1 tan A2 + tan A1 tan A3 + …
= the sum of the tangents taken two at a time,
S3 = tan A1 tan A2 tan A3 + tan A2 tan A3 tan A4 +…
= the sum of the tangents taken three at a time, and so on.
If A1=A2 = … = An=A, then
S1 = n tan A, S2 = nC2 tan2 A, S3 = nC3 tan3 A, … Therefore,
7. sin nA = cosn A (nC1 tan A- nC3 tan3 A + nC5 tan5 A - …)
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8. cos nA = cosn A (1- nC2 tan2 A + nC4 tan4 A +…)
9. tan n A =... A tan C -A tan C A tan C1
... A tan C A tan C -A tan 6
64
42
2
55
n33
n1
nnn
nC
10. sin n A + cos n A = cosn A (1 + nC1 tan A – nC2 tan2 A)
- nC3 tan3 A + nC4 tan4 A + nC5 tan5 A - nC6 tan6 A – nC7 tan7 A + …)
11. sin n A + cos n A = cosn A (-1 + nC1 tan A + nC2 tan2 A
- nC3 tan3 A - nC4 tan4 A + nC5 tan5 A + nC6 tan6 A …) =
2sin
2/sin
2)1(sin
n
13. cos + cos (+)+ cos (+2 )+…+ cos (+(n-1)) =
2/sin
2sin
2)1(cos
n
AQT1.8 VALUES OF TRIGONOMETRICAL RATIOS OF SOME IMPORTANT
ANGLES AND SOME IMPORTANT RESULTS
1. sin 15o = 22
13
2. cos 15o = 22
13
3. tan 15o = 2 - 3 = cot 75o
4. cot 15o = 2 + 3 = tan75o
5. sin 22
22
2
1
2
1o
6. cos 22
22
2
1
2
1o
7. tan 22 22
1o
=1
8. cot 22 22
1o
=1
9. sin 18o = 4
15 = cos 72o
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10. cos 18o =4
5210 = sin 72o
11. sin 36o = 4
5210 = cos 54o
12. cos 36o = 4
15 = sin 54o
13. sin 9o = 4
5553 = cos 81o
14. cos 9o = 4
5553 = sin 81o
15. sin sin (60o-) sin (60o+)= 4
1sin 3
16. cos cos (60o-) cos (60o+)= 4
1cos 3
17. tan tan(60o-) tan (60o+)= tan 3
18. cos 36o – cos 72o = 2
1
19. cos 36o cos 72o = 4
1
AQT1.9 EXPRESSIONS OF sin A/2 and cos A/2 IN TERMS OF SIN A
We have 2
2cos
2sin
AA= 1 + sin A and
2
2cos
2sin
AA= 1 - sin A
so that sin AAA
sin12
cos2
sin AAA
sin12
cos2
By adding and subtracting, we have
2 sin AAA
sin1sin12
= …(i)
and 2 cos AAA
sin1sin12
= …(ii)
In each of the formulae (i) and (ii) there are two ambiguous signs. To find these
ambiguities we proceed as follows:
We have
Sin
42sin2
2cos
2
AAA
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The RHS of this equation is positive if 2 n < 4
+
2
A< (2 n+1)
i.e. if 2n - 4
<
2
A< 2n +
4
3
hence sin2
A+ cos
2
Ais positive if
2
A lies between 2n -
4
and 2n +
4
3and it is
negative otherwise.
Similarly , sin 2
A- cos
2
A is positive if
2
A lies between
2n + 4
and 2n +
4
5 and otherwise it is negative.
These results can be shown graphically as given in the following figure:
AQT1.10 MAXIMUM AND MINIMUM VALUES OF
TRIGONOMETRICAL FUNCTIONS
As we have discussed in article AQT1.3 that – 1 sin x 1, - 1 cosx 1, -∞<tan
x<∞, |sec x| 1 and |cosec x| 1.
If there is a trigonometrical function of the form a sin x+b cos x, then by putting
a=r cos , b=r sin , we have
a sin x+b cos x = r cos sin x+r sin cos x
= r sin (x + ), where r = ,22 ba tan = a
b
Since – 1 sin (x+) 1 for all values of x. Therefore – r r sin (x+) r for all x
- 22 ba a sin x + b cos x 22 ba for all x.
Hence the maximum and minimum values of a trigonometrical function of the form
a sin x+b cos x are 22 ba and - 22 ba respectively.
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EXAMPLE 1 Find the maximum and minimum values of 6 sin x cos x+ 4 cos 2x.
SOLUTION We have 6 sin x cos x+ 4 cos 2x = 3 sin 2x + 4 cos 2x. Therefore the
maximum and minimum values of 3 sin 2x + 4 cos 2x are 22 43 and 22 43
i.e. 5 and -5 respectively.
EXAMPLE 2 PROVE THAT -45 COS +3 COS
3
+3 10 for all values of .
SOLUTION We have 5 cos + 3 cos
3
= 5 cos + 3 cos cos /3-3 sin sin
/3 =2
33cos
2
13 sin .
Since - sin2
33cos
2
13
2
33
2
1322
22
2
33
2
13
forall
-72
13cos -
2
33sin 7 for all
-75 cos +3cos (+/3) 7 for all
-7+3 5 cos +3 cos (+/3) +3 7+3 for all
-45 cos +3 cos (+/3) +3 10 for all .
EXAMPLE 3 Prove that
2222 )(2
1)(
2
1-x cos cx cos x sin sin a cabcabx for all x.
SOLUTION We have,
a sin2 x+b sin x cos x+c cos2 x =
2
2cos12sin
22
2cos1 xCx
bx
= 2
1[(a+c)+(c-a) cos 2x +b sin 2x]
- 22)( bac (c-a) cos 2x + b sin 2x 22)( bac
-2
1 22)( bac 2
1(c-a) cos 2x+
2
bsin 2x
2
1 22)( bac for all x
-2
1(a+c)-
2
1 22)( bca 2
1(a+c) +
2
1(c-a) cos 2x
+2
1b sin 2x
2
1(a-c)+
2
1 22)( bca for all x
2
1(a+c)-
2
1 22)( bca a sin2 x+b sin x cos x + c cos2 x 2
1(a-c)+
2
1
22)( bca for all x
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2222 )(2
1)(
2
1coscossinsin bcacaxcsxxbxa for all x
EXERCISE-AQTE.1
Mark the correct alternative(s) in each of the following:
1. The value of cos 10o – sin 10o is
(a) Positive (b) negative (c) 0 (d) 1
2. The value of cos 1o cos 2o cos 3o …cos 179o is
(a) 2
1 (b) 0 (c) 1 (d) none of these
3. The value of tan 1o tan 2o tan 3o … tan 89o is
(a) 1 (b) 0 (c) ∞ (d) 2
1
4. The maximum and minimum values of a cos 2 + b sin 2 are
(a) 2222 baandba (b) a+b and a-b
(c) a2 + b2 and – (a2 + b2) (d) none of these
5. Which of the following is correct
(a) sin 1o > sin 1(b) sin1o < sin 1(c) sin 1o = sin 1(d) sin1o = 180
sin 1
6. Given A=sin2 +cos4 , then for all real
(a) 1 A 2 (b) 4
3 A 1 (c)
16
13A 1 (d)
16
13
4
3 A
7. The value of o
o
15tan1
15tan12
2
is
(a) 1 (b) 3 (c) 2
3 (d) 2
8. If tan = - 4/3, then sin is
(a) -4/5 but not 4/5 (b) -4/5 or 4/5
(c) 4/5 but not – 4/5 (d) none of these
9. If sin +cos = 2 cos then cos -sin is equal to
(a) 2 cos (b) 2 sin (c) 2 (cos + sin ) (d) none of these
10. In a right angled triangle, the hypotenuse is four times as long as the
perpendicular drawn to it from the opposite vertex. One of the acute angle
is
(a) 15o (b) 30o (c) 45o (d) none of these
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11. If the angle is in the third quadrant and tan =2, then sin is equal to
(a) 5
2 (b)
5
2 (c)
2
5 (d) none of the these
12. The equation a sin x+b cos x=c, where |c|> 22 ba has
(a) a unique soln (b) Infinite no. of solns.
(c) no soln (d) none of these
13. Suppose that sin3 x sin 3x=
n
m
mc0
cos m x is an identity in x, where c1, c2,
c3 …, cn are constants and cn ≠ 0. Then the value of n is
(a) 4 (b) 5 (c) 6 (d) 8
14. In a triangle ABC, sin A-cos B=cos C, then angle B is
(a) /2 (b) /3 (c) /4 (d) /6
15. If lies in the first quadrant which of the following is not true
(a)
2tan
2
(b)
2sin
2
(c) cos2
2
<sin (d) sin
22
2
16. cos 2 + 2 cos is always
(a) greater than 2
3 (b) less than or equal to
2
3
(c) greater than or equal to 2
3 (d) none of these
17. If the interior angles of a polygon are in A.P. with common difference 5o
and the smallest angle 120o, then the number of sides of the polygon is
(a) 9 or 16 (b) 9 (c) 13 (d) 16
18. The maximum value of 5 cos + 3 cos
3
+ 3 is
(a) 5 (b) 10 (c) 11 (d) -11
19. The value of 16 sin 144o sin 108o sin 72o sin 36o is equal to
(a) 5 (b) 4 (c) 3 (d) 1
20. If A=tan 6o tan 42o and B=cot 66o cot 78o, then
(a) A=2B (b) A=3
1 (c) A=B (d) 3A=2B
21. If sin x+cosec x=2, then sin n x+cosen x is equal to
(a) 2 (b) 2n (c) 2n-1 (d) 2n-2
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22. If ,0sinsincoscos22
ba
then
(a) tan tan = )(
)(222
222
bya
axb
(b) x2 + y2 = a2 + b2
(c) tan tan =2
2
b
a (d)none of these
23. The values of lying between 0 and /2 and satisfying the equation
are 0
sin1cossin
4sin4cos1sin
4sin4cossin1
422
22
22
(a) 24
11
24
7 and (b)
24
5
24
7 and (c)
2424
5 and (d) none of
these.
24. The value of 3 cot 20o – 4 cos 20o is
(a) 1 (b) -1 (c) 0 (d) none of these.
25. The value of 3 cosec 20o – sec 20o is equal to
(a) 2 (b) 1 (c) 4 (d) -4
26. The equation sin2 = ,2
22
xy
yx is possible if
(a) x=y (b) x= - y (c) 2x = y (d) none of these
27. The value of sin (-) sin (-) cossec1 is equal to
(a) -1 (b) 0 (c) sin (d) none of these
28. If ,)(
)(
ba
ba
yx
yx
then
y
x
tan
tanis equal to
(a)a
b (b)
b
a (c) ab (d) none of these
29. If sin x+sin2 x=1, then value of cos2 x+cos4 x is
(a) 1 (b) 2 (c) 1.5 (d) none of these
30. |sin x+cos x|
(a) 2 (b) 2 (c) 2 (d) 2
1
31. If cos A= ,4
3then 32 sin
2
5sin
2
AA=
(a) 7 (b) 8 (c) 11 (d) none of these
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32. The value of
8
7cos1
8
5cos1
8
3cos1
8cos1
is
(a) 2
1 (b) cos /8 (c)
8
1 (d)
22
21
33. If tan2 =2 tan2 + 1, then cos 2 + sin2 equals
(a) -1 (b) 0 (c) 1 (d) none of these
34. If sin 2 =cos 3 and is an acute angle, then sin equals
(a)4
15 (b)
4
15 (c)
4
15 (d)
4
15
35. If y=sec2 +cos2 , ≠ 0, then
(a) y=0 (b) y2 (c) y-2 (d) y≠2
36. The value of
sin 14
13sin
14
11sin
14
9sin
14
7sin
14
5sin
14
3sin
14
is
(a)16
1 (b)
64
1 (c)
128
1 (d) none of these
37. The value of sin 14
5sin
14
3sin
14
is
(a) 1/16 (b) 1/8 (c) 1/2 (d) none of these.
38. If sin (+) =1, sin (-)= ½; [0, /2], then than (+2 ) tan (2 +)
is equal to
(a) 1 (b) -1 (c) 0 (d) none of these.
39. If cos (-)=a, cos (-)=b, then sin2 (-)+2ab cos (-)=
(a) a2 + b2 (b) a2 – b2 (c) b2-a2 (d) –a2 – b2
40. The value of sin
18
7sin
18
5sin
18
is
(a) 2
1 (b)
4
1 (c)
8
1 (d)
16
1.
41. The value of log tan 1o + log tan 2o + … + log tan 89o is
(a) 0 (b) -1 (c) 1 (d) ∞
42. If 1+sin x+sin2 x + sin3 x +…+…∞ is equal to 4+2 ,3 0< x < , then x=
(a) 6
(b)
4
(c)
63
or (d)
3
2
3
or
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43. If x cos +y sin =2a, x cos +y sin =2a and 2 sin 2
sin2
=1, then
(a) cos + cos =22
2
yx
ax
(b) cos cos =
22
222
yx
ya
(c) y2 = 4a (-x) (d) cos + cos =2 cos cos
44. If tan x= ,2
ca
b
a ≠ c; and y=a cos2 x+2 b sin x cos x+c sin2 x
z=a sin2 x-2 b sin x cos x+c cos2 x, then
(a)y=z (b) y+z=a-c (c) y-z=a-c (d) (y-z)=(a-c)2 +4 b2
45. If ++=2 , then
(a) 2
tan2
tan2
tan2
tan2
tan2
tan
(b) 2
tan2
tan2
tan2
tan2
tan2
tan
(c) 2
tan2
tan2
tan2
tan2
tan2
tan
(d) .02
tan2
tan2
tan2
tan2
tan2
tan
46. If sin -cos < 0, then lies between
(a) ,4
and4
3
nn n Z
(b) ,4
3 and
4
3
nn n Z
(c) ,4
2 and4
32
nn n Z
(d) ,4
2 and4
32
nn n Z
47. If 2 sin 2
then ,sin1sin12
AAA
A lies between
(a) ,4
3 2 and
42
nn n Z
(b) ,4
2 and4
3 2
nn n Z
(c) ,4
2 and4
32
nn n Z
(d) -∞ and +∞
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48. If 2 cos 2
then ,sin1sin12
AAA
A lies between,
(a) 4
3 2 and
4 2
nn
(b) 4
2 and4
2
nn
(c) 4
2 and4
3 2
nn
(d) -∞ and + ∞
49. The angle whose cosine equals to its tangent is given by
(a) cos =2 cos 18o (b) cos =2 sin 18o
(c) sin = 2 sin 18o (d) sin = 2 cos 18o
50. The value of cos 15
14cos
15
8cos
15
4cos
12
2 is
(a) 1 (b) 1/2 (c) 1/4 (d) 1/16.
51. The value of cos 15
7cos
15
6cos
15
5cos
15
4cos
15
3cos
15
2cos
15
is
(a) 66
1 (b)
77
1 (c)
82
1 (d) none of these.
52. The value of tan 5 is
(a) 42
53
tan5tan101
θtanθtan 10 - θ tan 5
(b)
42
53
tan5tan 101
θtanθtan 10 θ tan 5
(c) 42
35
tan5tan101
tanθθtan 10 - θ tan 5
(d) none of these.
53. If ,2
1)(
2
1coscos sin b sin a 22 kcac then k2 is equal to
(a) b2 + (a-c)2 (b) a2 + (b-c)2 (c) c2 + (a-b)2 (d) none of
these.
54. If ,sinsinsincos 22 k
then the value of k is
(a) 2cos1 (b) 2sin1 (c) 2sin2 (d) 2cos2
55. The value of sin 10o + sin 20o + sin 20o +…………+sin 360o is
(a) 1 (b) 0 (c) -1 (d) none of these.
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M C A E N T R A N C E | I I T J E E | F O U N D A T I O N S
56. The expression 3
)3(sin
2
3sin 44
-2
)5(sin
2sin 66
is
equal to
(a) 0 (b) 1 (c) 3 (d) sin 4 + cos 6
57. If A+B= ,4
then (tan A+1) (tan B+1) is equal to
(a) 1 (b) 2 (c) 3 (d) none of these.
58. If sin A+sin B=a and cos A+cos B=b, then cos (A+B)
(a) 22
22
ab
ba
(b)
22
2
ba
ab
(c)
22
22
ba
ab
(d)
22
22
ba
ba
59. If an angle is divided into two parts a and B such that A-B=x and tan
A:tan B=k:1, then the value of sin x is
(a)1
1
k
ksin (b)
1k
ksin (c)
1
1
k
ksin (d) none of
these.
60. The value of the expression 3 (sin-cos )4 +6 (sin +cos )2 +
4 (sin6 + cos6 ) is
(a)1 (b) -1 (c) 13 (d) 0.
61. If tan 2
5
2
and tan ,
4
3
2
then value of cos (+) is
(a) 725
364 (b)
725
627 (c)
339
240 (d) none of these.
62. If , , ,
2,0
then the value of
sinsinsin
)sin(
is
(a) < 1 (b) > 1 (c) = 1 (d) none of these.
63. If sin x+ sin y=3 (cos y –cos x), then the value of y
x
3sin
3sinis
(a) 1 (b) -1 (c) 0 (d) none of these.
64. If cos x=tan y, cos y=tan z, cos z=tan x, then the value of sin x is
(a) 2 cos 18o (b) cos 18o (c) sin 18o (d) 2 sin 18o
65. If k=sin6 x + cos6 x, then k belongs to the interval
(a)
4
5,
8
7 (b)
8
5,
2
1 (c)
1,
4
1 (d) none of these.
66. The value of tan 9o –tan27o –tan 63o + tan 81o is
(a) 2 (b) 3 (c) 4 (d) none of these.
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67. If tan2 tan2 + tan2 tan2 + tan2 tan2 +2 tan2 tan2 tan2 =1, then
the value of sin2 + sin2 +sin2 is
(a) 0 (b) 1 (c) 1 (d) none of these.
68. The value of oooo
e89tanlog...3tanlog2tanlog1tanlog 10101010 is
(a) 0 (b) e (c) 1/e (d) none of these.
69. For what and only what values of lying between 0 and is the inequality
sin cos3 > sin3 cos valid ?
(a)
4,0
(b)
2,0
(c)
2,
4
(d) none of
these.
70. If (sec A-tan A) (sec B-tan B) (sec C-tan C) = (sec A + tan A) (sec B+tan
B) (sec C+tan C) then each side is equal to
(a) 0 (b) 1 (c) -1 (d) 1
71. If << ,2
3then the expression
24cos42sinsin4 224
is equal to
(a) 2+4 sin (b) 2-4 sin (c) 2 (d) none of these.
72. If is an acute angle and sin ,2
1
2 x
x
then tan is
(a) 1
1
x
x (b)
1
1
x
x (c) 12 x (d) 12 x
73. The value of tan 2
182
o
is
(a) 5432 (b) )12)(23(
(c) )12)(23( (d) none of these
74. The value of cot 36o cot 72o tan 66o tan 78o is
(a) 1 (b) 2
1 (c)
4
1 (d)
8
1
75. The value of cot 36o cot 72o is
(a)5
1 (b)
5
1 (c) 1 (d) none of these
76. The value of cos 7
7cos
7
6cos
7
5cos
7
4cos
7
3cos
7
2cos
7
is
(a) 1 (b) -1 (c) 0 (d) none of these
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M C A E N T R A N C E | I I T J E E | F O U N D A T I O N S
77. The value of cos 7
6cos
7
4cos
7
2 is
(a) 1 (b) -1 (c) 2
1 (d)
2
1
78. The value of cos 7
3cos
7
2cos
7cos
is
(a)8
1 (b)
8
1 (c) 1 (d) 0
79. The value of cos 9
4cos
9
3cos
9
2cos
9
is
(a)8
1 (b)
16
1 (c)
64
1 (d) none of these
80. The value of cosec2 7
3cos
7
2cos
7
22 eec is
(a) 20 (b) 2 (c) 22 (d) 23
81. The value of sin 12o sin 48o sin 48o sin 54o is
(a) 1/4 (b) 1/8 (c) 1/16 (d) none of these
82. The value of sin 7
3sin
7
2sin
7
is
(a)cot14
(b)
7
3sin
7
2sin
7
is
(c) tan 14
(d)
14tan
2
1
83. tan6 9
tan279
tan339
24 =
(a) 0 (b) 3 (c) 3 (d) 9
84. A
A
A
A2
2
2
2
cos
3cos
sin
3sin =
(a) cos 2 A (b) 8 cos 2A (c) 1/8 cos 2A (d) none of these
85. If sin 625
336A where 450o < A < 540o, then sin
4
A=
(a)5
3 (b)
5
3 (c)
5
4 (d)
5
4
86. If y= ,3tan
tan
x
xthen
(a)
3,
3
1y (b)
3,
3
1y (c) y
3
1,3 (d)
3
1,3y y
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87. The value of cot2 9
4cot
9
2cot
9
22 is
(a) 0 (b) 3 (c) 9 (d) none of these
88. The value of sin 7
3sin
15
2 is
(a)8
1 (b)
8
7 (c)
2
7 (d) none of these
89. The value of sin 7
8sin
7
4sin
7
2 is
(a)8
7 (b)
8
1 (c)
2
7 (d)
2
7
90. The value of cos 15
16cos
15
5cos
15
4
15
2 coc is
(a) 0 (b) 1 (c) -1 (d) 8
1
91. If sin A + cos A= m and sin3 A + cos3 A=n, then
(a) m3 – 3m+n=0 (b) n3 -3n+2m=0
(c) m3 – 3m+2n=0 (d) m3 +3m+2n=0
92. If cos A+cos B= m and sin A + sin B=n where m, n≠0, then sin (A+B) is
equal to
(a)22 nm
mn
(b)
22
2
nm
mn
(c)
mn
nm
2
22 (d)
nm
mn
93. If 0 < A <6
and sin A+cos ,
2
7then tan
2
A=
(a)3
27 (b)
3
27 (c)
3
7 (d) none of these
94. The value of cos 11
9cos
11
7cos
11
5cos
11
3cos
11
(a) 0 (b) 2
1 (c)
2
1 (d) none of these
95. If 4n =, then the value of tan tan 3 tan 4 … tan (2n-2) tan (2n-
1) is
(a)0 (b) 1 (c) -1 (d) none of these
96. tan 9o – tan 27o – tan 63o + tan 81o is equal to
(a)0 (b) 1 (c) -1 (d) 4
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M C A E N T R A N C E | I I T J E E | F O U N D A T I O N S
97. For x R, tan x+22 2
tan2
1
2tan
2
1 xx + … +
11 2
tan2
1nn
xis equal to
(a) 2 cot 2x -
11 2
cot2
1nn
x (b)
11 2
cot2
1nn
x- 2 cot 2x
(c)
12
cotn
x- cot 2x (d) none of these
98. If A
Athenk
A
A
sin
3sin,
tan
3tan is equal to
(a) ,1
2
k
kkR (b)
3,
3
1,
1
2k
k
k
(c)
3,
3
1,
1
2k
k
k (d)
3 ,
3
1,
2
1k
k
k
99. If y = ,tansec
tansec2
2
then
(a)3
1<y<3 (b)
3 ,
3
1y
(c) -3 < y<-3
1 (d) none of these
100. If cos A = tan B, cos B=tan C, cos C=tan A, then sin A is equal to
(a) sin 18o (b) 2 sin 18o (c) 2 cos 18o (d) 2 cos 36o
101. If A1 A2A3A4a5 be a regular pentagon inscribed in a unit circle. Then (A1 A2)
(A1 A2) is equal to
(a) 1 (b) 3 (c) 4 (d) 5
102. If tan equals the integral solution of the inequality 4x2 -16x + 15<0 and
cos equals to the slope of the bisector of the first quadrant, then sin
(+) sin (-) is equal to
(a) 5
3 (b)
5
3 (c)
5
2 (d)
5
4
103. If
3
2cos
3
2cos
cos
zyx, then x+y+z=
(a) 1 (b) 0 (c) -1 (d) none of these
104. If cos A = ,4
3then the value of sin
2
5sin
2
AAis
(a) 32
1 (b)
8
11 (c)
32
11 (d)
16
11
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105. The minimum value of 9 tan2 +4cot2 is
(a) 13 (b) 9 (c) 6 (d) 12
106. If x1, x2, x3 …, xn are in A.P. whose common difference is , then the value
of sin (sec x1 sec x2 + sec x2 sec x3 + … + sec xn-1 sec xn) is
(a) nxx
n
coscos
)1sin(
1
(b)
n1 x coscosx
n sin
(c) sin (n-1) cos x1 cos xn
(d) sin n cos x1 cos xn.
107. If a sin2 x + b cos2 x=c, b sin2 y + a cos2 y = d and a tan x=b tan y, then
2
2
b
ais equal to
(a) ))((
))((
acad
bdcb
(b)
))((
))((
bdcb
acda
(c) ))((
))((
bdcb
acad
(d)
))((
))((
daca
dbcb
108. If an+1 =
to ... a a a
a-1 cos then ),1(
2
1
321
20
na is equal to
(a) 1 (b) -1 (c) a0 (d) 1/a0
109. If , , , are the smallest positive angles in ascending order of
magnitude which have their sines equal to the positive quantity k, then
the value of 4 sin 2
sin 2
sin 22
is equal to
(a) k12 (b) k12 (c) 2
1 k (d) none of these.
110. The value of cos y cos
yx
2 cos
2
cos x + sin y cos
yx
2 sin x cos
2
is zero if
(a) x=0 (b) y=0 (c) x=y+4
(d) x= y
4
3
111. If cos x-sin cot sin x=cos , then tan 2
xis equal to
(a) 2
tan2
cot
(b)2
cot2
cot
(c) 2
tan2
tan
(d)2
cot2
cot
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112. The expression cos2 A cot2 A-sec2 A tan2 A – (cot2 A-tan2 A) (sec2 A cosec2
A-1) is equal to
(a) 1 (b) -1 (c) 0 (d) none of these
113. If sin +cos =m, then sin6 +cos6 is equal to
(a)4
)1(34 22 m (b)
4
)1(34 22 m
(c)4
)1(43 22 m (d) none of these
114. If 0 x and ,30818181222 coscossin xxx then x is equal to
(a)6
(b)
3
(c)
6
5 (d)
3
2
115. If cos (A-B)=5
3and tan A tan B=2, then
(a) cos A cos B = 5
1 (b) sin A sin B = -
5
2
(c) cos (A-B) = - 5
1 (d) none of these
116. The value of oo
oo
16cot76cot
)16cot76cot3(
is
(a) cot44o (b) tan 44o (c) tan2o (d) cot 46o
117. If sin x+sin2 x=1, then cos8 x+2 cos6 x+cos4 x =
(a) 0 (b) -1 (c) 2 (d) 1
118. If x=y cos 3
2= z cos ,
3
4then xy + yz + 2x =
(a) -1 (b) 0 (c) 1 (d) 2
119. The values of (0< <360o) satisfying cosec + 2=0 are
(a) 210o, 300o (b) 240o, 300o (c) 210o, 240o (d) 210o , 310o
120. If sin = sin and cos =cos , then
(a) 02
sin a
(b) 02
cos
(c) 02
sin
(d) 02
cos
121. If sin 1 + sin 2 + sin 3 = 3, then cos 1 + cos 2 + cos 3 =
(a) 3 (b) 2 (c) 1 (d) 0
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122. If A lies in the third quadrant and 3 tan A-4=0, then 5 sin 2A + 3 sin A +
4 cos A =
(a) 0 (b) 5
24 (c)
5
24 (d)
5
48
123. tan 5x tan 3x tan 2x =
(a) tan 5x-tan 3x-tan 2x (b) 2x cos-3x cos-5x cos
2x sin-3x sin-5x sin
(c) 0 (d) none of these
124. sin 12o sin 24o sin 48o sin 84o =
(a) cos 20o cos 40o cos 60o cos 80o
(b) sin 20o sin 40o sin 60o sin 80o
(c) 3/15 (d) none of these
125. If A + B+C = ,2
3then cos 2 A+cos 2 B + cos 2 C =
(a) 1-4 cos A cos B cos C (b) 4 sin A sin B sin C
(c) 1+2 cos A cos B cos C (d) 1-4 sin A sin B sin C
126. If A+C=B, then tan A tan B tan C =
(a) tan tan B+tan C (b) tan B-tan C-tan A
(c) tan A+tan C-tan B (d) – (tan A tan B+tan C)
127. sin 75o + cos 75o =
(a) 3 /2 (b) 2/3 (c) 1/ 2 (d) 1/2
128. If tan = ,b
athen b cos 2 +a sin 2 =
(a) a (b) b (c) b/a (d) none of these
129. tan 15o =
(a) 1/3 (b) 3 -2 (c) 2- 3 (d) none of these
130.
8cos1
8
3cos1
8
5cos1
8
7cos1
is equal to
(a)2
1 (b) cos
8
(c)
8
1 (d)
22
21
131. If A=cos2 +sin4 , then for all values of ,
(a) 1 A 2 (b) 116
13 A
(c) 16
13
4
3 A (d) 1
4
3 A
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132. The minimum value of the expression sin +sin + sin , where , ,
are real numbers satisfying ++= is
(a) positive (b) zero (c) negative (d) -3
133. Which of the following statement is incorrect?
(a) sin = - 1/5 (b) cos =1 (c) sec =2
1 (d) tan =20
134. The value of sin 14
7 sin
14
5 sin
14
3 sin
14
is
(a) 1 (b) 1/4 (c) 1/8 (d) 7/2
135. If sin + cosec = 2, then sin2 + cosec2 is equal to
(a) 1 (b) 4 (c) 2 (d) none of these
136. If tan 2
1 and tan ,
3
1then the value of + is
(a) /6 (b) (c) zero (d) /4
137. If sin x+sin2 x=1, then the value of
cos12 x+3 cos10 x+3 cos8 x+cos6 x + 2 cos4 x + cos2 x-2 is equal
(a) 0 (b) 1 (c) 2 (d) none of these
138. The maximum value of 12 sin -9 sin2 is
(a) 3 (b) 4 (c) 5 (d) none of these
139. If f(x)=cos2 x+sec2 x, its value always is
(a) f(x) < 1 (b) f(x) = 1 (c) 2 > f(x)> 1 (d) f(x) 2.
140. The maximum value of 3 cos x+4 sin x+5 is
(a) 5 (b) 9 (c) 7 (d) none of these
141. Maximum value of a cos +b sin is
(a) a+b (b) a-b (c) |a|+|b| (d) 22 ba
142. Maximum value of 3 cos +4 sin is
(a) 3 (b) 4 (c) 5 (d) none of these
143. The maximum value of sin (x+/6)+ cos (x+/6) in the interval (0, /2) is
attained at
(a) /12 (b) /6 (c) /3 (d) /2
144. If A+B+C = (A, B, C>0) and the angle C is obtuse, then
(a) tan A tan B>1 (b) tan A tan B<1
(c) tan A tan B=1 (d) none of these
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145. If A, B, C are acute positive angles such that A+B+C= and cot A cot B
cot C=K, then
(a) 33
1k (b)
33
1k (c)
9
1k (d)
3
1k
146. If cos +cos =0 = sin +sin , then cos 2 +cos 2 =
(a) -2 sin (+) (b) -2 cos (+) (c) 2 sin (+) (d) -2 cos (+)
147. If sin is the GM between sin and cos , then cos 2 =
(a)
4 sin 2 2 (b)
4 cos 2 2 (c)
4 cos 2 2 (d)
4 sin 2 2
148. tan 15 tan
5
2 tan 3
15 tan
5
2 is equal to
(a) - 3 (b) 1/ 3 (c) 1 (d) 3
149. If OP makes 4 revolutions in one second, the angular velocity in radians per
second is
(a) (b) 2 (c) 4 (d) 8
150. If tan = ,sin
cos1
then
(a) tan 3 = tan 2 (b) tan 2 =tan
(c) tan 2 = tan (d) none of these
151. The value of cot 72
1o
+ tan 672
1o
- cot 672
1o
- tan 72
1o
is
(a) 3 (2+ 2 ) (b) 2 (3+ 3 ) (c) 3+ 2 (d) 2+ 3
152. If sin =-3/5 and lies in the third quadrant, then the value of cos (/2) is
(a) 5
1 (b)
10
1 (c)
5
1 (d)
10
1
153. The value of 2
2tan
2cot
xx(1-2 tan x cot 2 x) is
(a) 1 (b) 2 (c) 3 (d) 4
154. tan sin
2cos
2=
(a) 1 (b) -1 (c) 2
1sin 2 (d) none of these
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155. sin2
9
4sin
18
7sin
9sin
18
222 =
(a) 1 (b) 2 (c) 4 (d) none of these
156. If 5 sin =3 sin (+2 )≠0, then tan (+) is equal to
(a) 2 tan (b) 3 tan (c) 4 tan (d) 6 tan
157. The value of tan 20o + 4 sin 20o is
(a) 3 (b) 2 (c) 3
1 (d) none of these
158. Given << ,2
3then the expression
24cos4sinsin4 224
is equal
to
(a) 2 (b) 2+4 sin (c) 2-4 sin (d) none of these
159. sin
10
13sin
10
=
(a) 2
1 (b)
2
1 (c)
4
1 (d) 1
160. tan 20
9tan
20
7tan
20
5tan
20
3tan
20
=
(a) 1 (b) -1 (c) 2
1 (d) none of these
161. sin 20o sin 40o in 60o sin 80o =
(a) 16
3 (b)
16
5 (c)
16
3 (d)
16
5
162. Given that (1+ )11(tan)1 xxx . Then sin 4x =
(a) 4x (b) 2x (c) x (d) none of these
163. The value of oo
oo
16cot76cot
16cot76cot3
is
(a) cot 44o (b) tan 44o (c) tan 2o (d) cot 46o
164. tan 3A-tan 2A-tan A =
(a) tan 3 A tan 2 A tan A (b) – tan 3 A tan 2 A tan A
(c) tan A tan 2 A – tarr 2 A tan 3 A-tan 3 A tan A (d) none of these
165. cos 70o – cos 10o =
(a) 2
1 (b) cos 40o (c) – sin 40o (d) none of these
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166. If A+B+C = 180o, then CBA
CBA
tantantan
tantantan =
(a) tan A tan B tan C (b) 0 (c) 1 (d) none of these
167. If A+B+C = 180o, then cos 2A+cos 2B +cos2C=
(a) 1+4 cos A cos B sin C (b) -1+4 sin A sin B cos C
(c) 1-4 cos A cos B cos C (d) none of these
168. If A+B+C = 180o, then sin 2A + sin 2B+Sin 2C =
(a) 4 cos A cos B cos C (b) 4 sin A sin B sin C
(c) 2 sin A sin B sin C (d) 8 sin A sin B sin C
169. cos 35o+ cos85o+cos 155o =
(a) 0 (b) 3
1 (c)
2
1 (d) cos 275o
170. If A= sin8 +cos14 , then for all values of ,
(a) A1 (b) 0<A<1 (c) 1<2A3 (d) none of these
171. The value of sin 50o – sin 70o + sin 10o is equal to
(a) 1 (b) 0 (c) 2
1 (d) 2
172. If a 3 cos x+5 sin
6
x b for all x, then
(a) a = - 34,34 b (b) a = - 19,19 b
(c) a = - 38,38 b (d) none of these
173. For real values of
(a) cos (cos ) < 0 (b) cos (cos ) > 0
(c) cos (cos ) 0 (d) none of these
174. sec2 =2)(
4
yx
xy
is true if and only if
(a) x+y≠0 (b) x=y, x≠0 (c) x = y (d) x≠0, y≠0
175. tan 7 ½o =
(a) 13
)31(22
(b)
31
31
(c) 3
3
1 (d) 2 2 + 3
176. The value of cos 225o + sin 165o is
(a) 0 (b) 2
13 (c)
2
13 (d)
2
12
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M C A E N T R A N C E | I I T J E E | F O U N D A T I O N S
177. The value of cos 2/15 cos 4/15 cos 8/15 cos 14 /15 is
(a) 1/16 (b) 1/8 (c) 3/4 (d) 1/12
178. If is an acute angle and tan = ,7
1then the value of
22
22
seccos
seccos
ec
eis
(a) 3/4 (b) 1/2 (c) 2 (d) 5/4
179. The value of sin2 5o + sin2 10o + sin2 15o +…+ sin2 85o+sin2 90o is
(a) 7 (b) 8 (c) 9 (d) 10
180. sin2 /8+sin2 /9+sin2 7 /18+sin29
4=
(a) 1 (b) 4 (c) 2 (d) 0
1 a 26 a 51 b 76 b 101 d 126 b 151 b 176 a
2 b 27 a 52 a 77 d 102 c 127 b 152 b 177 a
3 a 28 b 53 a 78 a 103 b 128 b 153 d 178 a
4 a 29 a 54 b 79 b 104 c 129 c 154 d 179 c
5 b 30 c 55 b 80 d 105 d 130 c 155 b 180 c
6 b 31 c 56 b 81 b 106 a 131 d 156 c A V A Q U A N T A N S K E Y T R I G O N O M E T R Y U N I T 1
7 c 32 c 57 b 82 b 107 b 132 b 157 a
8 b 33 b 58 b 83 c 108 c 133 c 158 a
9 b 34 a 59 c 84 b 109 b 134 c 159 c
10 a 35 d 60 c 85 c 110 d 135 c 160 a
11 b 36 b 61 b 86 b 111 b,c 136 d 161 c
12 c 37 b 62 a 87 b 112 c 137 d 162 c
13 c 38 a 63 b 88 b 113 a 138 b 163 a
14 a 39 a 64 d 89 c 114 a,b 139 d 164 a
15 b 40 c 65 c 90 b 115 a 140 d 165 c
16 a 41 a 66 c 91 c 116 a 141 d 166 c
17 b 42 d 67 c 92 b 117 d 142 c 167 c
18 b 43 c 68 d 93 a 118 b 143 a 168 b
19 a 44 c 69 a 94 c 119 d 144 b 169 a
20 c 45 a 70 d 95 b 120 c 145 a 170 b
21 a 46 d 71 c 96 d 121 d 146 b 171 b
22 a,b 47 a 72 c 97 b 122 a 147 a,c 172 b
23 a 48 b 73 a 98 c 123 a 148 d 173 b
24 a 49 c 74 a 99 a 124 a 149 d 174 b
25 c 50 d 75 b 100 b 125 d 150 b 175 a