# Solution of Chapter 5. Trigonometric Ratios (RD Sharma - Mathematics Book)

## Exercise 5.1

1

In each of the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

(ix) (x)

(xi) (xii)

2

In a , right angled at B, AB = 24 cm, BC = 7 cm. then find the other ratios.

3

In Fig. 5.37, find tan P and cot R. Is tan P = cot R?

4

If , compute cos A and tan A.

5

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

6

In , right angled at Q, PQ = 4 cm and RQ = 3 cm. Find the values of sin P, sin R, sec P and sec R.

7

If , evaluate:

(i)

(ii)

8

If 3 cot A = 4, check whether or not.

9

If , find the value of

10

If 3 tanθ = 4, find the value of

11

If 3 cot = 2, find the value of

12

If , prove that

13

If , show that

14

If , show that sin (1-tan) =

15

If cot = , show that

16

If tan = , show that

17

If , find the value of

18

If , find the value of

19

If , find the value of

20

If , find the value of

21

If , find the value of

22

If, find the value of

23

If , verify that

24

If , prove that

25

If sec A = , verify that

26

If , prove that

27

If , find that .

28

If , find in terms of a and b.

29

If 8 tan A = 15, find

30

If , find

31

If , show that

32

If , find the value of

33

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

34

If are acute angles such that tan A = tan P, then show that .

35

In a , right angled at A, if tan C = , find the value of sin B cos C + cos B sin C.

36

(i) The value of tan A is always less than (ii) 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) for some angle .

## Exercise 5.2

1

Evaluate each of the following:

2

Evaluate each of the following:

3

Evaluate each of the following:

4

Evaluate each of the following:

5

Evaluate each of the following:

6

Evaluate each of the following:

7

Evaluate each of the following:

8

Evaluate each of the following:

9

Evaluate each of the following:

10

Evaluate each of the following:

11

Evaluate each of the following:

12

Evaluate each of the following:

13

Evaluate each of the following:

14

Evaluate each of the following:

15

Evaluate each of the following:

16

Evaluate each of the following:

17

Evaluate each of the following:

18

Evaluate each of the following:

19

Evaluate each of the following:

20

Find the value of x in each of the following:

21

Find the value of x in each of the following:

22

Find the value of x in each of the following:

23

Find the value of x in each of the following:

24

Find the value of x in each of the following:

25

Find the value of x in each of the following:

26

If , verify that:

(i)

(ii)

(iii)

(iv)

27

If , verify that

(i)

(ii)

(iii)

28

If A = 30° and B = 60°, verify that

(i) .

(ii)

29

If sin (A - B) = sin A cos B – cos A sin B and cos (A - B) = cos A cos B + sin A sin B, find the values of sin 15° and cos 15°.

30

In a right triangle ABC, right angled at C, if = 60° and AB = 15 units. Find the remaining angles and sides.

31

Ifis a right triangle such that = 90°, = 45° and BC = 7 units. Find , AB and AC.

32

In a rectangle ABCD, AB = 20 cm, = 60°, calculate side BC and diagonals AC and BD.

33

If sin (A + B) = 1 and cos (A – B) = 1, 0° < A + B ≤90°, A ≥ B find A and B.

34

If tan (A – B) = and tan (A + B) = , 0° < A + B ≤90°, A > B find A and B.

35

If sin (A – B) = and cos (A + B) = , 0° < A + B ≤90°, A < B find A and B.

36

In a right angled at B, . Find the values of

(i)

(ii)

37

Find acute angles A and B, if sin (A + 2B) = and cos (A + 4B) = 0, A > B.

38

If A and B ae acute angles such that tan A = , tan B = and tan (A + B) = , find A + B.

39

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

## Exercise 5.3

1

Evaluate the following:

(i) (ii)

(iii) (iv)

(v)

2

Evaluate the following:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii) cosec 31° - sec 59°

(viii)

(ix)

(x)

(xi)

3

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

4

Express in terms of angles between 0° and 30°.

5

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

6

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

(i)

(ii)

7

Prove that:

(i)

(ii)

(iii)

(iv)

8

Prove the following:

(i)

(ii)

(iii)

(iv)

(v)

9

Evaluate:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

10

In where are acute angles, find the degree measure of .

11

If A, B, C are the interior angles of a , show that:

(i) (ii)

12

If are acute angles, find the degree measure of satisfying

13

If is a positive acute angle such that , find the value of .

14

If , where and are acute angles, find the value of .

15

If , where and are acute angles, find the value of .

16

If , where is an acute angle, find the value of A.

17

If , where is an acute angle, find the value of A.

## CCE - Formative Assessment

1

Write the maximum and minimum values of sin θ.

2

Write the maximum and minimum values of cos θ.

3

What is the maximum value of ?

4

What is the maximum value of ?

5

If, find the value of

6

If , find the value of

7

If 3cot θ = 4, find the value of

8

Given , what is the value of ?

9

If, rite the value of

10

If tan A = 3/4 and A + B = 90°, then what is the value of cot B ?

11

If A + B = 90° and cos B = 3/5, what is the value of sin A?

12

Write the acute angle θ satisfying √3 sin θ = cos θ

13

Write the value of cos 1° cos 2° cos 3°…. cos 179° cos 180°.

14

Write the value of tan 10° tan 15° tan 75° tan 80°.

15

If A + B = 90° and tan A = 3/4, what is cot B?

16

If tan A = 5/12, find the value of (sin A + cos A) sec A.

1

If θ is an acute angle such that cos θ = 3/5, then =

2

If tan θ = a/b then is equal to

3

If 5 tan θ – 4 = 0, then the value of is

4

If 16 cot x = 12, then equals

5

If 8 tan x = 15, then sin x - cos x is equal to

6

If, then =

7

If tan θ = 3/4, then cos2 θ – sin2 θ =

8

If θ is an acute angle such that tan2 θ = 8/7, then the value of is

9

If 3cos θ = 5 sin θ, then the value of is

10

If tan2 45° - cos2 30° = x sin 45° cos 45°, then x =

11

The value of cos2 17° - sin2 73° is

12

The value of is

13

If = tan260° – tan230° then

14

If A and B are complementary angles, then

15

If x sin (90° - 0) cot (90° - θ) = cos (90° - θ), then x =

16

If x tan 45° cos 60° = sin 60° cot 60°, then x is equal to

17

If angles A, B, C of a ΔABC form an increasing AP, then sin B =

18

If θ is an acute angle such that sec2θ = 3, then the value of is

19

The value of tan 1° tan 2° tan 3° ……tan 89° is

20

The value of cos 1° cos 2° cos 3°…… cos 180° is

21

The value of tan 10° tan 15° tan 75o tan 80. is

22

The value of is

23

If θ and 2θ – 45° are acute angles such that sin θ = cos (2θ – 45°), then tan θ is equal to

24

If 5θ and 4θ are acute angles satisfying sin 5θ = cos 4θ, then 2 sin 3θ – √3 tan 4θ is equal to

25

If A + B = 90°, then is equal to

26

is equal to

27

is equal to

28

sin 2A = 2 sin A is true when A =

29

is equal to

30

If A, B and C are interior angles of a triangle ABC, then =.

31

If cos θ = 2/3 then 2 sec2θ + 2tan2θ – 4 is equal to

32

tan 5° × tan 30° × 4 tan 85° is equal to

33

The value of + cot1° cot 2° cot 3°.... cot 90°, is