Solutions of chapter 9. Values of Trigonometric Functions at Multiples and Submultiple of a | RD Sharma - Mathematics - Class 11

RD Sharma - Mathematics

Book: RD Sharma - Mathematics

Chapter: 9. Values of Trigonometric Functions at Multiples and Submultiple of a

Subject: Maths - Class 11^th

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Chapter Exercises

Exercise 9.1

Exercise 9.2

Exercise 9.3

Very Short Answer

MCQ

Exercise 9.1

1

Prove the following identities:

2

Prove the following identities:

3

Prove the following identities:

4

Prove the following identities:

5

Prove the following identities:

6

Prove the following identities:

7

Prove the following identities:

8

Prove the following identities:

9

Prove the following identities:

10

Prove the following identities:

11

Prove the following identities:

(cos α + cos β)² + (sin α + sin β)² = 4 cos²

12

Prove the following identities:

13

Prove the following identities:

1 + cos² 2x = 2(cos⁴ x + sin⁴ x)

14

Prove the following identities:

cos³ 2x + 3 cos 2x = 4(cos⁶ x – sin⁶ x)

15

Prove the following identities:

(sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

16

Prove the following identities:

17

Prove the following identities:

cos 4x = 1 – 8 cos² x + 8 cos⁴ x

18

Prove the following identities:

sin 4x = 4 sin x cos³ x – 4 cos x sin³ x

19

Prove the following identities:

3(sin x – cos x)⁴+6(sin x + cos x)²+4(sin⁶ x + cos⁶ x) = 13

20

Prove the following identities:

2(sin⁶ x + cos⁶ x) – 3(sin⁴ x + cos⁴ x)+ 1 =0

21

Prove the following identities:

cos⁶ x – sin⁶ x = cos 2x

22

Prove the following identities:

23

Prove the following identities:

cot² x – tan² x = 4 cot 2x cosec 2x

24

Prove the following identities:

cos 4x – cos 4α = 8(cos x – cos α) (cos x + cos α) (cos x – sin α) (cos x + sin α)

25

Prove the following identities:

sin 3x + sin 2x – sin x = 4 sin x cos

26

Prove that:

27

Prove that:

28

If and x lies in the IIIrd quadrant, find the values of and sin 2x.

28

If and x lies in the IInd quadrant, find the values of sin 2x and

29

If and x lies in IInd quadrant, find the values of and

30

0 ≤ x ≤ π and x lies in the IInd quadrant such that Find the values of and

30

If and x is acute, find tan 2x.

30

If and find the value of sin 4x.

31

If then find the value of

32

If and show that cos 2A = sin 4B

33

Prove that:

34

Prove that:

35

Prove that:

36

Prove that:

37

If 2 tan α = 3 tan β, prove that tan (α - β)

38

If sin α + sin β = a and cos α + cos β = b, prove that

38

If sin α + sin β = a and cos α + cos β = b, prove that

39

If prove that

40

If prove that

41

If sec (x + α) + sec(x - α) = 2 sec x, prove that cos x

42

If and prove that

43

If and prove that

44

If a cos 2x + b sin 2x = c has α and β as its roots, then prove that

44

If a cos 2x + b sin 2x = c has α and β as its roots, then prove that

44

If a cos 2x + b sin 2x = c has α and β as its roots, then prove that

45

If cos α + cos β = 0 = sin α + sin β, then prove that cos 2α + cos 2β = - 2 cos (α + β).

Exercise 9.2

1

Prove that:

sin 5x = 5 sin x – 20 sin³ x + 16 sin⁵ x

2

Prove that:

4(cos³10^o + sin³20^o) = 3 (cos10^o + sin20^o)

3

Prove that:

4

Prove that:

5

Prove that:

6

Prove that:

7

Prove that:

8

Prove that:

sin 5x = 5 cos⁴x sin x – 10 cos²x sin³ x + sin⁵ x

9

Prove that:

10

Prove that:

For all values of x.

11

Prove that:

for all values of x

Exercise 9.3

1

Prove that:

2

Prove that:

3

Prove that:

4

Prove that:

5

Prove that:

6

Prove that:

7

Prove that:

8

Prove that:

9

Prove that:

10

Prove that:

Very Short Answer

1

If then write the value of k.

2

If then write the value of m sin x + n cos x.

3

If then write the value of

4

If then write the value of in the simplest form.

5

If then write the value of

6

If then write the value of

7

In a right-angled triangle ABC, write the value of sin² A + sin² B + sin² C.

8

Write the value of cos² 76^o + cos² 16^o – cos 76^o cos 16^o.

9

If then write the value of

10

Write the value of

11

If then find the value of tan 2A.

12

If sin x + cos x = a, find the value of sin⁶ x + cos⁶ x.

13

If sin x + cos x = a, find the value of |sin x – cos x|

MCQ

1

Mark the Correct alternative in the following:

is equal to

2

Mark the Correct alternative in the following:

is equal to

3

Mark the Correct alternative in the following:

The value of is

4

Mark the Correct alternative in the following:

If cos 2x + 2 cos x = 1 then, (2 – cos² x) sin² x is equal to

5

Mark the Correct alternative in the following:

For all real values of x, cot x – 2 cot 2x is equal to

6

Mark the Correct alternative in the following:

The value of is

7

Mark the Correct alternative in the following:

If in a ∆ABC, tan A + tan B + tan C = 0, then cot A cot B cot C =-

8

Mark the Correct alternative in the following:

If and then λ =

9

Mark the Correct alternative in the following:

If 2 tan α = 3 tan β, then tan (α - β) =

10

Mark the Correct alternative in the following:

If then

11

Mark the Correct alternative in the following:

If sin α + sin β = a and cos α – cos β = b, then

12

Mark the Correct alternative in the following:

The value of is

13

Mark the Correct alternative in the following:

The value of is

14

Mark the Correct alternative in the following:

The value of is

15

Mark the Correct alternative in the following:

If 5 sin α = 3 sin (α + 2 β) ≠ 0, then tan (α + β) is equal to

16

Mark the Correct alternative in the following:

The value of 2 cos x – cos 3x – cos 5x – 16 cos³ x sin² x is

17

Mark the Correct alternative in the following:

If A = 2 sin² x – cos 2x, then A lies in the interval

18

Mark the Correct alternative in the following:

The value of is equal to

19

Mark the Correct alternative in the following:

If tan (/4 + x) + tan (/4 – x) = λ sec 2x, then

20

Mark the Correct alternative in the following:

The value of is

21

Mark the Correct alternative in the following:

is equal to

22

Mark the Correct alternative in the following:

The value of 2 sin² B + 4 cos (A + B) sin A sin B + cos 2 (A + B) is

23

Mark the Correct alternative in the following:

The value of is

24

Mark the Correct alternative in the following:

2(1 – 2 sin² 7x) sin 3x is equal to

25

Mark the Correct alternative in the following:

If α and β are acute angles satisfying then tan α =

26

Mark the Correct alternative in the following:

If then cos α =

27

Mark the Correct alternative in the following:

If (2ⁿ + 1) x = π, then 2ⁿ cos x cos 2x cos2² x ….. cos 2^{n – 1} x =

28

Mark the Correct alternative in the following:

If tan x = t then tan 2x + sec 2x is equal to

29

Mark the Correct alternative in the following:

The value of cos⁴ x + sin⁴ x – 6 cos² x sin² x is

30

Mark the Correct alternative in the following:

The value of cos (36^o – A) cos (36^o + A) + cos(54^o – A) cos (54^o + A) is

31

Mark the Correct alternative in the following:

The value of is

32

Mark the Correct alternative in the following:

The value of is

33

Mark the Correct alternative in the following:

The value of is

34

Mark the Correct alternative in the following:

is equal to

35

Mark the Correct alternative in the following:

If n = 1, 2, 3, …., then cos α cos 2 α cos 4 α … cos 2^{n – 1} α is equal to

36

Mark the Correct alternative in the following:

If then b cos 2x + a sin 2x is equal to

37

Mark the Correct alternative in the following:

If then cos 2α is equal to

38

Mark the Correct alternative in the following:

The value of cos² 48^o – sin² 12^o is