5 Introduction to Trigonometry - Study Point · Pythagoras theorem Hypotenuse 22= Perpendicular +...

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NTSE, NSO Diploma, XI Entrance

Introduction to TrigonometryCLASS - XMathematics

CHAPTER

5We are Starting from a Point but want to Make it a Circle of Infinite Radius

Introduction to Trigonometry

Trigonometry:-The branch of mathematics that deals with the relations between the sides and

angles of plane or spherical triangles, and the calculations based on them.

PAB is a right-angled triangle in which PAB = . This side PB is opposite to angle and its

called opposite side (perpendicular); AB is the adjacent side (base) in relation to angle ;

AP is the hypotenuse.

Pythagoras theorem

Hypotenuse2 = Perpendicular

2 + Base

2

QP2 = ER

2 + QR

2

sine, cosine, tangent of an angle

(i) The ratio AP

PBis called the sine of angle and, in short form, it is written as sin.

sin = Hypotenuse

deOppositesi

AP

PB

(ii) The ratio AP

ABis called the cosine of angle and is briefly written as cos .

cos = Hypotenuse

sideAdjacent

AP

AB

(iii) The ratio AB

PB is called the tangent of angle and is briefly written as tan .

tan = sideAdjacent

sideOposite

AB

PB

Note: (1) sin , cos , and tan depend only on the angle and not on the size of the right

triangle used to measure them.

(2) ‘sin ’ is an abbreviation for ‘sine of angle ’, it is not the product of sin and .

A B

P

P

Q R

Q R

Hypotenuse

Base

Perpendicular

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Other Trigonometric Ratios

Besides three trigonometric ratios, sine, cosine and the tangent of an angle , there are three

other trigonometric ratios, namely,

cosecant (abbreviated as cosec), secant (abbreviated as sec) and cotangent (abbreviated as cot)

of the angle . We define them as follows.

Cosec =

sin

1

PB

AP

angletooppositeSide

hypotenuse

Sec =

cos

1

AB

AP

angletoadjacentSide

Hypotenuse

Cot =

tan

1

BP

AB

angletooppositeSide

angletoadjacentSide.

Relation between sin , cos and tan

The trigonometrical ratios sin , cos and tan of an angle are closely related.

Where RQP =

sin=QP

PR

cos=QP

QR

tan= QR

PR

Now PQ/QR

PQ/PR

cos

sin

=

QR

PRtan .

If any one of these is known, the other two can be easily calculated.

so tan =

sin

coscot,

cos

sin

Trigonometry Identities

sin cos tan

=

P B P

cosec sec cot H H B

P

Q R

Q R

P

Q R

Q R

Hypotenuse

(H)

Base (B)

Perpendicular (P)

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Exercise 5.1

1. In ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :

(i) sin A, cos A

(ii) sin C, cos C

2. In figure find tan P – cot R.

3. In triangle ABC, right-angled at B, if tan A = 3

1, find the value of:

(i) sin A cos C + cos A sin C

(ii) cos A cos C – sin A sin C

4. In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.

5. In ABC , right angled at C, if tan A = 1

3, and tan B = 3 , prove that (sin A cos B + cos A sin B) = 1

6. In OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine the values of sin Q and cos Q.

7. In PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and

tan P.

8. If B and Q are acute angles such that sin B = sin Q, then prove that B = Q.

9. If A and B are acute angles such that cos A = cos B, then show that A = B.

10. Given tan A = 3

4, find the other trigonometric ratios of the angle A.

11. Given 15 cot A = 8, find sin A and sec A.

12. If 8 tan A = 15, find sin A – cos A

13. Given sec = 12

13, calculate all other trigonometric ratios.

14. If cot = 8

7, evaluate :

(i) )cos1)(cos1(

)sin1()sin1(

(ii) cot

2

15. Consider ACB, right-angled at C, in which AB = 29

units, BC = 21 units and ABC = . Determine the

values of

(i) cos2 + sin

2

(ii) cos2 – sin

2

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16. If 3 cot A = 4, check whether A

A2

2

tan1

tan1

= cos

2 A – sin

2 A or not.

17. If tan = 1

7, show that

2 2

2 2

(cos s ) 3

4(cos s )

ec ec

ec ec

.

18. If tan = b

a find the value of

sincos

sincos

19. If tan = b

a , then

cossin

cossin

ba

ba

is equal to

22

22

ba

ba

20. If sin = 22

22

ba

ba

, find the values of other five trigonometric ratios.

21. If sin A = 3

1, evaluate cos A coses A + tan A sec A.

22. If cosec A = 2, find the value of A

A

A cos1

sin

tan

1

23. If 3 cos - 4 sin = 2cos + sin , find tan

24. If tan = 3

4 then the value of

sin1

sin1

is

25. State whether the following are true or false. Justify your answer.

(i) The value of tan A is always less than 1.

(ii) sec A = 5

12 for some value of angle A.

(iii) cos A is the abbreviation used for the cosecant of angle A.

(iv) cot A is the product of cot and A.

(v) sin3

4 for some angle .

26. If tan 2tan

1

, find the value of

2

2

tan

1tan .

27. In the figure, CD AB. If AD = 4 units, DB = 6 units and CD = 3 units, find the values of

(i) sin A,

(ii) cot A

(iii) sec B

(iv) tan B.

28. In the figure ABC is a right angled at B. BSC is right angle at S, and BRS is right angled at R, AB

= 18 cm, BC = 7.5 cm, RS = 5 cm BSR = x0 and SAB = y

0. Find

(i) tan x0

(ii) sin y0

(iii) cos y0

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29. In the figure B = 900, BC = CD =

2

a, BAC = 0 and AD = b. Find

(i) sin (ii) cos

30. Using the measurement given in the adjoining figure, find

(i) cos

(ii) tan

(iii) cosec

31. If )(

2tan

22 nm

mn

, find the values of other t-ratios of .

32. If 7 sin A = 24 cos A, prove that 14 tan A + 25 cos A – 7 sec A = 30.

Table of values of Trigonometric Ratios

Angle

T-

Ratio

0 30 45 60 90

sin 0 1 / 2 1 / 2 3 / 2 1

cos 1 3 / 2 1 / 2 1 / 2 0

tan 0 1 / 3 1 3 Not defined

cot Not defined 3 1 1 / 3 0

sec 1 2 / 3 2 2 Not defined

cosec Not defined 2 2 2 / 3 1

In this table the value of trigonometric ratio is defined only for first quadrant

For first quadrant the following points are given below

The value of sin or cos never exceeds 1, but the value of sec and cosec

is always greater than or equal to 1.

The value of sin increases as increases and decreases as decreases

The value of cos decreases as increases and increases as decreases

The value of tan increases as increases and decreases as decreases

The value of cot decreases as increases and increases as decreases

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Exercise 5.2

1. In ABC, right-angled at B, AB = 5 cm and ACB = 30°. Determine the lengths of the sides BC and

AC.

2. In PQR, right-angled at Q, PQ = 3 cm and PR = 6 cm. Determine QPR and PRQ.

3. Evaluate the following :

(i) sin 60° cos 30° + sin 30° cos 60°

(ii) 2 tan2 45° + cos

2 30° – sin

2 60°

(iii) 00

0

30cos30sec

45cos

ec

(iv) o2o2

o2o2

45cos45sec

60tan60sin

(v) ooo

o2o2o2

45tan30cos30sin2

30tan445cos30sin5

4. Choose the correct option and justify your choice :

(i) 02

0

30tan1

30tan2

(A) sin 600 (B) cos 60

0 (C) tan 60

0 (D) sin 30

0

(ii) 02

02

45tan1

45tan1

(A) tan 900 (B) 1 (C) sin 45

0 (D) 0

(iii) sin 2A = 2 sin A is true when A =

(A) 00 (B) 30

0 (C) 45

0 (D) 60

0

(iv) o

o

30tan1

30tan22

=

(A) cos 60° (B) sin 60

0 (C) tan 60

0 (D) sin 30°

(v) The value of oo

oo

36tan60tan1

30tan60tan

is equal to.

(A) sin 30o (B) cos 30

o (C) cot 30

o (D) tan 30

o

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5. If = 30o then

2

2

tan1

tan1is equal to.

(A) sin 2 (B) cos 2 (C) tan2 (D) 1

6. If A= 30o then 4Cos

3 A – 3cos A is

(A) sin 3A (B) sin 2A (C) cos 3A (D) cos A

7. If tan (A + B) = 3 and tan (A – B) = 3

1; 0° < A + B

8. If sin (A – B) = 2

1, cos (A + B) =

2

1, 0° < A + B

9. If sin (A + B) = 1 and cos (A – B) = 3

2, find A and B.

10. Show that :

(i) cos 600 cos 30

0 – sin 60

0 sin 30

0 = cos 90

0

(ii) cos 600 = 1 – 2sin

2 30

0 = 2cos

2 30

0 – 1

(iii) 0 0

0 0

tan 60 tan 30

1 tan 60 tan 30

= tan 30

0

11. Taking = 300, verify each of the following :

(i) sin 2 = 2sin cos

(ii) cos 2 = 2cos2 - 1 = 1 – 2sin

2

(iii) sin 3 = 3sin - 4sin3

(iv) cos 3 = 4cos3 - 3cos

12. tan 2 = 2

2 tan

1 tan

.

13. Taking A = 600 and B = 30

0, verify that sin (A – B) = sin A cos B – cos A sin B.

14. If A = B = 450, verify that

(i) sin (A + B) = sin A cos B + cos A sin B

(ii) sin (A – B) = sin A cos B – cos A sin B

(iii) cos (A + B) = cos A cos B – sin A sin B

(iv) cos (A – B) = cos A cos B + sin A sin B

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15. If A = 60 and B = 300, verify that

(i) sin (A + B) = sin A cos B + cos A sin B

(ii) sin (A – B) = sin A cos B – cos A sin B

(iii) cos (A + B) = cos A cos B – sin A sin B

(iv) cos (A – B) = cos A cos B + sin A sin B

(v) tan (A – B) = tan tan

1 tan tan

A B

A B

16. Find the value of x in each of the following :

(i) 2 sin 3x = 3

(ii) 2 sin 12

x

(iii) 3 sin x = cos x

(iv) tan x = sin 450 cos45

0 + sin30

0

(v) 3 tan 2x = cos600 + sin45

0 cos45

0

(vi) cos2x = cos600 cos30

0 + sin60

0 sin30

0

(vii) Cos ( 40 + x ) = sin 30 , find x

17. State whether the following are true or false. Justify your answer. 900

(i) sin (A + B) = sin A + sin B.

(ii) The value of sin

(iii) The value of cos

(iv) sin .

(v) cot A is not defined for A = 0°.

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Trigonometric Ratio of Complementary angles

Sin(90 - ) = Cos Cos(90 - ) = sin

tan (90 - ) = cot cot (90 - ) = tan

cosec (90 - ) = sec sec (90 - ) = cosec

In the first quadrant (I), all ratios are positive.

In the second quadrant (II), sine (and cosec) are positive.

In the third quadrant (III), tan (and cotan) are positive.

In the fourth quadrant (IV), cos (and sec) are positive.

Exercise 5.3

Evaluate Angles

1. Evaluate 0

0

25cot

65tan.

2. Evaluate :

(i) 0

0

72cos

18sin

(ii) 0

0

64cot

26tan

(iii) cos 480 – sin 42

0

(iv) cosec 310 – sec 59

0

3. Show that :

(i) tan 48° tan 23° tan 42° tan 67° = 1

(ii) cos 38° cos 52° – sin 38° sin 52° = 0

Evaluate

4.

5. -

6. + - 4 cos

2 45.

7. +

8. - - cos 0 + tan 15 tan 25 tan 60 tan 65 tan 75

9. cos (40 + ) – sin (50 - ) +

cos2 20 + cos

2 70

sin2 59 + sin 31

sin 27

cos 63

cos 63

sin 27

2

sin 47

cos 43

cos 43

sin 47

cos 70

sin 20

cos 55 cosec 35

tan 5 tan 25 tan 45 tan 65 tan 85

2 cos 67

sin 23

tan 40

cot 50

cos2 40 + cos

2 50

sin2 40 + sin

2 50

2

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10. cos2 20 + + sin 35 sec 55

Find the value of

11. - - sin 90.

12. + - 8 sin

2 30.

13. Evaluate: sin cos - -

14. Prove that: + = 2 cosec .

15. Evaluate: + +

16. Show that: tan1 tan 2 tan 3 tan 87 tan 88 tan 89 = 1.

16. Show that: cos10 cos2

0 cos3

0 --- cos100

0 = 0

16. Show that: sin 00 sin 1

0 sin 3

0 sin 30

0 sin 60

0 sin 90

0 = 0

17. Evaluate without using tables:

cot tan (90 - ) – sec (90 - ) cosec + sin2 25 + sin

2 65 + 3

18. Without using trigonometric tables, evaluate:

(i) 0 0

0 0

0 0

tan 50 sec50cos 40 cosec50

cot 40 sec 40co

(ii) 0002

0202

0202

26sin64cos64sin70sin20sin

70cos20cos

(iii) sec2 10

0 – cot

2 80

0 +

)90cos(sin)90sin(cos

75sin15cos15sin00

0002

19. Prove that

(i) sin (500 + ) – cos (40

0 - ) = 0

(ii) coses (650 + ) – sec (25

0 - ) = 0

20. Without using trigonometric tables, evaluate the following :

0 0 0

0

cot (90 ) . sin (90 ) cot 40

sin tan50

- (cos

2 20

0 + cos

2 70

0)


)20tan70(cos

38sin52sec30tan2

)42cos48(cos4

tan)90(cos

0202

020202

0202

202

ec

ec

cos2 70

sin2 59 + sin

2 31

2 cos 67

sin 23

tan 40

cot 50

cos 70

sin 20

cos 59

sin31

sincos (90 - ) cos

sec (90 - )

cos sin (90 - ) sin

Cosec (90 - )

cos (90 - )

1 + sin (90 - )

1 + sin (90 - )

cos (90 - )

sincos . sin (90 - )

cos (90 - )

cos . sin . cos (90 - )

sin (90 - )

sin2 27 + sin

2 63

cos2 40 + cos

2 50

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0002

0202

022

32tan58cot58cos40cot50sec

70coscos

ec


0

0

39 2

sec51 3

sec

co . tan 17

0 tan 38

0 tan 60

0 tan 52

0 tan 73

0


2 20 0

0 0

tan 20 20

sec70 sec70

cot

co

+ 2 tan 15

0 tan 37

0 tan 53

0 tan 60

0 tan 75

0

25. Prove that

(i) sin (700 + ) – cos (20

0 - ) = 0 (ii) tan (55

0 + ) – cot (35

0 + ) = 0

26. Find , if sin ( + 360) = cos , where + 36

0 is an acute angle

27. If tan 2 = cot ( + 60), where 2 and + 6

0 are acute angles, find the value of .

28. If sin 5 = cos 4 , where 5 and 4 are acute angles, find the value of .

29. If sec 5A = cosec (A - 360), where 5A is an acute angle , find the value of A

30. If A,B,C are the interior angles of a triangle ABC, prove that

(i) tan 2

cot2

BAC

(ii) 2

cos2

sinAAB

(iii) If 2 + 450 and 30

0 - are acute angles, find the degree measure of satisfying sin (2 + 45

0)

= cos (300 - )

31. If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

32. Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

33. If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.

34. If tan A = cot B, prove that A + B = 90°.

35. If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.

36. Express sin 670 + cos 75

0 in terms of trigonometric ratios of angles between 0

0 and 45

0.

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Formula

sin2 + cos

2 = 1 => sin2 = 1 – cos

2 => cos2 = 1 – sin

2

1 + tan2 = sec

2 => sec2 - tan

2 = 1 => sec2 - 1 = tan

2

1 + cot2 = cosec

2 => cosec2 - cot

2 = 1 => cosec2 - 1 = cot

2

sin sin cos tan cot sec cosec

sin sin 2cos1

2tan1

tan

2cot1

1

sec

1sec2

eccos

1

cos 2sin1

cos

2tan1

1

2cot1

cot

sec

1

eccos

1eccos 2

tan

2sin1

sin

cos

cos1 2

tan

cot

1 1sec2

1eccos

1

2

cot

sin

sin1 2

2cos1

cos

tan

1

cot

1sec

1

2 1eccos 2

sec

2sin1

1

cos

1 2tan1

cot

cot1 2

sec

1eccos

eccos

2

cosec

sin

1

2cos1

1

tan

tan1 2

2cot1

1sec

sec

2

eccos

Exercise 5.4

1. Express the ratios cos A, tan A and sec A in terms of sin A.

2. Prove that sec A (1 – sin A) (sec A + tan A) = 1.

3. Prove that :AA

AA

coscot

coscot

=

1cos

1cos

Aec

Aec.

4. Prove that 1cossin

1cossin

=

tansec

1

using the identity sec

2 2 .

5. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

6. Write all the other trigonometric ratios of

7. Evaluate :

(i) 0202

0202

73cos17cos

27sin63sin

(ii) sin 25

0 cos 65

0 + cos 25

0 sin 65

0

8. Choose the correct option. Justify your choice :

(i) 9 sec2 A – 9 tan

2 A =

(A) 1 (B) 9 (C) 8 (D) 0

(ii) (1 + tan ) (1 + cot – cosec ) =

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(A) 0 (B) 1 (C) 2 (D) –1

(iii) (sec A + tan A) (1 – sin A) =

(A) sec A (B) sin A (C) cosec A (D) cos A

(iv) A

A2

2

cot1

tan1

=

(A) sec2

A (B) – 1 (C) cot2 A (D) tan

2 A

9. Prove the following identities, where the angles involved are acute angles for which the expressions are

defined.

(i) (cosec - cot )2 =

cos1

cos1

(ii) AA

A

A

Asec2

cos

sin1

sin1

cos

(iii) A

A

A

A

cos1

sin

sec

sec1 2

(iv) 1sincos

1sincos

AA

AA= cosec A + cot A, using the identity cosec

2 A = 1 + cot

2 A.

(v) A

A

sin1

sin1

= sec A + tan A

(vi)

tan

coscos2

sin2sin3

3

(vii) (sin A + cosec A)2 + (cos A + sec A)

2 = 7 + tan

2 A + cot

2 A

(viii) cosec A – sin A) (sec A – cos A) = AA cottan

1

(ix) AA

A

A

A 22

2

2

tancot1

tan1

cot1

tan1

10. 1 – cos2 - sin

2 = 0

11. sec 1 – sin2 = 1

12. (sec2 A – 1) (cosec

2 A – 1) = 1.

13. (1 + tan2) cos

2 = 1

14. (1 + tan2)(1 – sin )(1 + sin ) = 1.

15. Prove that:

(i) (sec2 - 1)(1 – cosec

2) = 1.

(ii) cosec2 + sec

2 = cosec

2 sec

2.

(iii) + 1. tan2 A =

16. + = 1 + tan + cot

cos2

A

sin2 A

1

cos2

A

cot

1 – tan

tan

1 – cot

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17. tan - cot =

18. (cos - sin )2 + (cos + sin )

2 = 2.

19. = (sec + tan )2

=

20. - = -

21. sec4 – sec

2 = tan

4 + tan

2.

22. If sin + sin2 = 1, prove that cos

2 + cos

4 = 1.

23. 1 + 1 + =

24. Prove that

(i) (sin2 A cos

2 B – cos

2 A sin

2 B) = (sin

2 A – sin

2 B)

(ii) (tan2 A sec

2 B – sec

2 A tan

2 B) = (tan

2 A – tan

2 B).

25. Prove that (tan2 A – tan

2 B) =

2 2 2 2

2 2 2 2

(sin sin ) (cos cos )

cos cos cos cos

A B B A

A B B A

.

26.

sin1

cos

1cos

1cos

ec

ec

27. 1sec

1sec

1sec

1sec

= 2 cosec .

28. 002 900,)cot(cos

cotcos

cotcos

ec

ec

ec= 1 + 2 cot2 + 2 cosec cot

29.

2

cos

sin1

tansec

tansec

30.

2

2

2

cot1

tan1

cot1

tan1

31. AAAA

A

A

Acossin

sincos

sin

tan1

cos 2

32. sec A (1 – sin A) (sec A + tan A) = 1

33. (sec A + cos A) (sec A – cos A) = tan2 A + sin

2 A

34. 1cot

tan12

2

= tan

2 , 450

2 sin2 - 1

sincos

sec + tan

sec - tan

1 + sin

cos

1

sec - tan

1

cos

1

cos

1

sec + tan

1

tan2

1

cot2

1

sin2 - sin

4

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35. 2cossin

1

sin

cos

cos

sin222

2

2

2

AAA

A

A

A

36. 1sec

1sec

sintan

sintan

37. cot + tan = cosec sec

38. 2)cot(coscos1

cos1

ec

39. 2)tan(secsin1

sin1

40. tan - cot =

cossin

1sin2 2 =

cossin

cos21 2

41.

cos1

sin

= cosec + cot

42. (1 + tan2 ) (1- sin ) (1 + sin ) = 1

43. xxxxxx tansec

1

cos

1

cos

1

tansec

1

44. (cosec - sin ) (sec - cos ) = cottan

1

45. A

A

AA

AA

cos

sin1

1sectan

1sectan

46. cotcos

1

sin

1

sin

1

cotcos

1

ecec

47.

cottan1

tan1

cot

cot1

tan

48. AAec

Aec

Aec

Aec 2tan221cos

cos

1cos

cos

49. If x = a cos and y = b sin , then b2x

2 + a

2y

2 =

(a) a2 b

2 (b) ab (c) a

4 b

4 (d) a

2+ b

2

50. If x = a sec and y = b tan , then b2 x

2 – a

2 y

2 =

(a) ab (b) a2 – b

2 (c) a

2 b

2 (d) a

2b

2

51. If a cot + b cosec = p and b cot + a cosec = q, then p2 – q

2 =

(a) a2 – b

2 (b) b

2 – a

2 (c) a

2 + b

2 (d) b – a

52. If a cos + b sin = m and a sin - b cos = n, then a2 + b

2 =

(a) m2 – n

2 (b) m

2 n

2 (c) n

2 – m

2 (d) m

2 + n

2

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MCQ

1. If is an acute angle such that cos = 2

3 sin tan 1, then

5 2 tan

(a) 16

625 (b)

1

36 (c)

3

160 (d)

160

3

2. If tan = a a sin bcos

, then is equal tob a sin bcos

(a) 2 2

2 2

a b

a b

(b)

2 2

2 2

a b

a b

(c)

a b

a b

(d)

a b

a b

3. If 5 tan - 4 = 0, then the value of 5sin 4cos

is5sin 4cos

(a) 5

3 (b)

5

6 (c) 0 (d)

1

6

4. If 16 cot x = 12, then sin x cos x

equalssin x cos x

(a) 1

7 (b)

3

7 (c)

2

7 (d) 0

5. If 8 tan x = 15, then sin x – cos x is equal to

(a) 8

17 (b)

17

7 (c)

1

17 (d)

7

17

6. If tan = 2 2

2 2

1 cosec sec, then

7 cosec sec

(a) 5

7 (b)

3

7 (c)

1

12 (d)

3

4

7. If tan 2 23

, then cos sin4

(a) 7

25 (b) 1 (c)

7

25

(d)

4

25

8. If is an acute angle such that 2 8tan

7 , then the value of

(1 sin ) (1 sin )is

(1 cos ) (1 cos )

(a) 7

8 (b)

8

7 (c)

7

4 (d)

64

49

9. If 3 cos = 5 sin , then the value of 3

3

5sin 2sec 2cosis

5sin 2sec 2cos

(a) 271

979 (b)

316

2937 (c)

1

2 (d) None of these

10. If tan2 45

o – cos

2 30

o = x sin 45

o cos 45

o, then x =

(a) 2 (b) -2 (c) 1

2 (d)

1

2

11. The value of cos2 17

o – sin

2 73

o is

(a) 1 (b) 1

3 (c) 0 (d) –1

12. The value of 3 o 3 o

3 o 3 o

cos 20 cos 70is

sin 70 sin 20

(a) 1

2 (b)

1

2 (c)1 (d) 2

13. If 2 o 2 o

2 o 2 o

2 o 2 o

x cosec 30 sec 45tan 60 tan 30 , then x

8cos 45 sin 60

(a) 1 (b) –1 (c) 2 (d) 0

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14. If A and B are complementary angles, then

(a) sin A = sin B (b) cos A = cos B

(c) tan A = tan B (d) sec A = cosec B

15. If x sin (90o ) cot (90

o ) = cos (90o ) = cos (90

o ), then x =

(a) 0 (b) 1 (c) 1 (d) 2

16. If x tan 45o cos 60

o = sin 60

o cot 60

o, then x is equal to

(a)1 (b) 3 (c)1

2 (d)

1

2

17. If angles A, B, C of a ABC form an increasing AP, then sin B =

(a) 1

2 (b)

3

2 (c) 1 (d)

1

2

18. If is an acute angle such that 2sec 3, then the value of 2 2

2 2

tan cosecis

tan cosec

(a) 4

7 (b)

3

7 (c)

2

7 (d)

1

7

19. The value of tan 1o tan 2

o tan 3

o ……. tan 89

o is

(a) 1 (b) –1 (c) 0 (d) None of these

20. The value of cos 1o cos 2

o cos 3

o…… cos 180

o is

(a) 1 (b) 0 (c) –1 (d) None of these

21. The value of tan 10o tan 15

o tan 75

o tan 80

o is

(a) –1 (b) 0 (c) 1 (d) None of these

22. The value of o o o

o o o

cos(90 )sec(90 ) tan tan(90 )

cotcosec(90 )sin(90 )cot(90 )

is

(a) 1 (b) –1 (c) 2 (d) –2

23. If and 2 -45o are acute angles such that sin = cos (2 -45

o), then tan is equal to

(a) 1 (b) –1 (c) 3 (d) 1

3

24. If 5 and 4 are acute angles satisfying sin 5 = cos 4 , then 2 sin 3 3 tan 3 is equal to

(a) 1 (b) 0 (c) –1 (d) 1+ 3

25. If A + B = 90o, then

2

2

tan A tan B tan Acot B sin Bisequal to

sin AsecB cos A

(a) cot2 A (b) cot

2 B (c) – tan

2 A (d) – cot

2 A

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Answers Exercise 5.1

1. (i) 25

7,

25

24 (ii)

25

24,

25

7 2. 0

3. (i) 1 (ii) 0

6 25

7,

25

24 7.

13

12,

13

5,

5

12

10. sin A = 4/5, cos A 3/5 cosec A = 5/4, sec A = 5/3 cot A = 3/4

11. 17

15,

8

17 12.

17

7

13. sin 13

5, cos =

13

12, tan =

12

5 14. (i)

64

49 (ii)

64

49

15. (i) 1 (ii) 41/841

16. yes

17. ab

ab

18.

22

22

ba

ba

19. cos = 22

224

ba

ba

, tan =

22

22

4 ba

ba cosec =

22

22

ba

ba

, sec =

22

22

4 ba

ba , cot =

22

22

4

4

ba

ba

20. 8

3216 21. 2 23.

5

124.

3

1

25. (i) False (ii) False (iii) False (iv) false (v) false

26. 2 27. (i) 5

3 (ii)

3

4 (iii)

2

5 (iv)

2

1

28. (i) 5

4 (ii)

13

5 (iii)

13

12 29. (i)

22 34

3

ab

a

(ii)

22

22

34

2

ab

ab

30. (i) 13

12 (ii)

9

12 (iii)

9

15 31.

22

2sin

nm

mn

,

22

22

cosnm

nm

Exercise 5.2

1. 35 cm , 10 cm 2. 600, 30

0

3. (i) 1, (ii) 2 (iii) 8

629 (iv)

2

5 (v)

6

5)32(

4. (i) (B) cos 600 (ii) (D) 0 (iii) (A) 0

0 (iv) (C) tan 60

0 (v) (D) tan 30

0

5. (B) cos2 6. (C) cos

3 A

7. 450, 15

0 8. 45, 15

9. 60, 30

16. (i) 200 (ii) 60

0 (iii) 30 (iv)45 (v) 15 (vi) 15 (vii) 200 17. (i) False (ii) True (iii) False (iv) False (v) True

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Exercise 5.3

1. 1 2. (i) 1 (ii)1 (iii) 0 (iv) 0 4. 1 5. 0

6. 0 7. 2 8. 3 9. 1 10. 2

11. 0 12 0 13. 0 15. 2 cos2 17. 3

18. (i) 2 (ii) 2 (iii) 2 20. 1 21. 12

5 22. 2

23. 1 + 2 3 26. 270

27. 280 28. 10

0

29. 21 30. (iii) 360

31. 290 32. tan 50 + sin 15

0

33. 360 35. 22

0 36. cos 23 + sin 15

Exercise 5.4

1. AA

AA

22

2

sin1

1,

sin1

sin,sin1

5.

AA

A

A cot

1,

cot

cot1,

cot1

1 2

2

6. 1sec

1,1sec,

sec

1,

sec

1sec

2

22

AA

AA

A,

1sec

sec

2 A

A

7. (i) (B)1 (ii) (C) 2 (iii) D cos A (iv) (D) tan2 A 8. (i) (B) 9 (ii) (C) 2 (iii) (D) cos A (iv) (D) tan

2 A

49. (a) 50. (d) 51. (b) 52. (d)

MCQ

1. c 2. a 3. c 4. a 5. d 6. d 7. a 8. a 9. a 10. a

11. c 12. c 13. a 14. d 15. b 16. a 17. b 18. d 19. a 20. b

21. c 22. c 23. a 24. b 25. b

5 Introduction to Trigonometry - Study Point · Pythagoras theorem Hypotenuse 22= Perpendicular +...

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