TRIGONOMETRIC RATIOS, IDENTITIES AND MAXIMUM ......TRIGONOMETRIC RATIOS, IDENTITIES AND MAXIMUM &...

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TRIGONOMETRY Page | 1 __________________________________________________________________________________ __________________________________________________________________________________ MCA ENTRANCE | IIT JEE | FOUNDATIONS 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 (90 o ) 1 o =60 minutes (60’) 1 o = 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 R D 2 90 1radian = 180 degrees = 57 o 17’ 45” (approximately) and 1 degree = 180 radian

Transcript of TRIGONOMETRIC RATIOS, IDENTITIES AND MAXIMUM ......TRIGONOMETRIC RATIOS, IDENTITIES AND MAXIMUM &...

Page 1: TRIGONOMETRIC RATIOS, IDENTITIES AND MAXIMUM ......TRIGONOMETRIC RATIOS, IDENTITIES AND MAXIMUM & MINIMUM VALUES OF TRIGONOMETRICAL EXPRESSIONS AQT1.1 INTRODUCTION DEFINITION: An angle

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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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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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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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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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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