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RD Chapter 6- Graphs of Trigonometric Functions Ex-6.1 Interview Questions Answers

Question 1 :
Sketch the graphs of the following functions:
(i) f (x) = 2 sin x, 0 ≤ x ≤ π
(ii) g (x) = 3 sin (x – π/4), 0 ≤ x ≤ 5π/4
(iii) h (x) = 2 sin 3x, 0 ≤ x ≤ 2π/3
(iv) ϕ (x) = 2 sin (2x – π/3), 0 ≤ x ≤ 7π/3
(v) Ψ (x) = 4 sin 3 (x – π/4), 0 ≤ x ≤ 2π
(vi) θ (x) = sin (x/2 – π/4), 0 ≤ x ≤ 4π
(vii) u (x) = sin2 x, 0 ≤ x ≤ 2π υ (x) = |sin x|, 0 ≤ x ≤ 2π
(viii) f (x) = 2 sin πx, 0 ≤ x ≤ 2

Answer 1 :

(i) f (x) = 2 sin x, 0 ≤ x ≤ π
We know that g (x) = sin x is a periodic function with period π.
So, f (x) = 2 sin x is a periodic function with period π. So, we will draw the graph of f (x) = 2 sin x in the interval [0, π]. The values of f (x) = 2 sin x at various points in [0, π] are listed in the following table:

x

0(A)

π/6 (B)

π/3 (C)

π/2 (D)

2π/3 (E)

5π/6 (F)

Π (G)

f (x) = 2 sin x

0

1

√3 = 1.73

2

3 = 1.73

1

0

The required curve is:

(ii) g (x) = 3 sin (x – π/4), 0 ≤ x ≤ 5π/4

We know that if f (x) isa periodic function with period T, then f (ax + b) is periodic with periodT/|a|.

So, g (x) = 3 sin (x –π/4) is a periodic function with period π. So, we will draw the graph of g (x)= 3 sin (x – π/4) in the interval [0, 5π/4]. The values of g (x) = 3 sin(x – π/4) at various points in [0, 5π/4] are listed in the following table:

x

0(A)

π/4 (B)

π/2 (C)

3π/4 (D)

π (E)

5π/4 (F)

g (x) = 3 sin (x – π/4) 

-3/√2 = -2.1

0

3/√2 = 2.12

3

3/√2 = 2.12

0

The required curve is:

(iii) h (x) = 2 sin 3x, 0 ≤ x ≤ 2π/3

We know that g (x) = sinx is a periodic function with period 2π.

So, h (x) = 2 sin 3x isa periodic function with period 2π/3. So, we will draw the graph of h (x) = 2sin 3x in the interval [0, 2π/3]. The values of h (x) = 2 sin 3x at variouspoints in [0, 2π/3] are listed in the following table:

x

0 (A)

π/6 (B)

π/3 (C)

π/2 (D)

2π/3 (E)

h (x) = 2 sin 3x

0

2

0

-2

0

The required curve is:

(iv) ϕ (x) = 2 sin (2x – π/3), 0 ≤ x ≤ 7π/3

We know that if f(x) isa periodic function with period T, then f (ax + b) is periodic with periodT/|a|.

So, ϕ (x) = 2 sin (2x – π/3) is a periodicfunction with period π. So, we will draw the graph of ϕ (x) = 2 sin (2x –π/3), in the interval [0, 7π/5]. The values of ϕ (x) = 2 sin (2x –π/3), at various points in [0, 7π/5] are listed in the following table:

x

0 (A)

π/6 (B)

2π/3 (C)

7π/6 (D)

7π/5 (E)

ϕ (x) = 2 sin (2x – π/3) 

-√3 = -1.73

0

0

0

1.98

The required curve is:

(v) Ψ (x) = 4 sin 3 (x – π/4), 0 ≤ x ≤ 2π

We know that if f(x) isa periodic function with period T, then f (ax + b) is periodic with periodT/|a|.

So, Ψ (x) = 4 sin 3(x – π/4) is a periodic function with period 2π. So, we will draw thegraph of Ψ (x) = 4 sin 3 (x – π/4) in the interval [0, 2π]. The values ofΨ (x) = 4 sin 3 (x – π/4) at various points in [0, 2π] are listed in thefollowing table:

x

0 (A)

π/4 (B)

π/2 (C)

π (D)

5π/4 (E)

2π (F)

Ψ (x) = 4 sin 3 (x – π/4) 

-2√2 = -2.82

0

2√2 = 2.82

0

1.98

-2√2 = -2.82

The required curve is:

(vi) θ (x) = sin (x/2 – π/4), 0 ≤ x ≤ 4π

We know that if f(x) isa periodic function with period T, then f (ax + b) is periodic with periodT/|a|.

So, θ (x) = sin (x/2 –π/4) is a periodic function with period 4π. So, we will draw the graphof θ (x) = sin (x/2 – π/4) in the interval [0, 4π]. The valuesof θ (x) = sin (x/2 – π/4) at various points in [0, 4π] are listed in thefollowing table:

x

0 (A)

π/2 (B)

π (C)

2π (D)

5π/2 (E)

3π (F)

4π (G)

θ (x) = sin (x/2 – π/4) 

-1/√2 = -0.7

0

1/√2 = 0.7

1/√2 = 0.7

0

-1/√2 = -0.7

-1/√2 = -0.7

The required curve is:

(vii) u (x) = sin2 x,0 ≤ x ≤ 2π υ (x) = |sin x|, 0 ≤ x ≤ 2π

We know that g (x) = sinx is a periodic function with period π.

So, u (x) = sin2 x is a periodic function withperiod 2π. So, we will draw the graph of u (x) = sin2 xin the interval [0, 2π]. The values of u (x) = sin2 xat various points in [0, 2π] are listed in the following table:

x

0 (A)

π/2 (B)

Π (C)

3π/2 (D)

2π (E)

u (x) = sin2 x

0

1

0

1

0

The required curve is:

(viii) f (x) = 2 sin πx, 0 ≤ x ≤ 2

We know that g (x) = sinx is a periodic function with period 2π.

So, f (x) = 2 sin πx isa periodic function with period 2. So, we will draw the graph of f (x) = 2 sinπx in the interval [0, 2]. The values of f (x) = 2 sin πx at various points in[0, 2] are listed in the following table:

x

0 (A)

1/2 (B)

1 (C)

3/2 (D)

2 (E)

f (x) = 2 sin πx

0

2

0

-2

0

The required curve is:

Question 2 :
Sketch the graphs of the following pairs of functions on the same axes:
(i) f (x) = sin x, g (x) = sin (x + π/4) 
(ii) f (x) = sin x, g (x) = sin 2x
(iii) f (x) = sin 2x, g (x) = 2 sin x
(iv) f (x) = sin x/2, g (x) = sin x

Answer 2 :

(i) f (x) = sin x, g (x) = sin (x + π/4) 
We know that the functions f (x) = sin x and g (x) = sin (x + π/4) are periodic functions with periods 2π and 7π/4.
The values of these functions are tabulated below:
Values of f (x) = sin x in [0, 2π]

x

0

π/2

π

3π/2

f (x) = sin x

0

1

0

-1

0

Values of g (x) = sin (x + π/4) in [0, 7π/4]

x

0

π/4

3π/4

5π/4

7π/4

g (x) = sin (x + π/4)

1/√2 = 0.7

1

0

-1

0

The required curve is:

(ii) f (x) = sin x, g (x) = sin 2x

We know that the functions f(x) = sin x and g (x) = sin 2x are periodic functions with periods 2π and π.

The values of these functions are tabulated below:

Values of f (x) = sin x in [0, 2π]

x

0

π/2

π

3π/2

f (x) = sin x

0

1

0

-1

0

Values of g (x) = sin (2x) in [0, π]

x

0

π/4

π/2

3π/4

π

5π/4

3π/2

7π/4

g (x) = sin (2x)

0

1

0

-1

0

1

0

-1

0

The required curve is:

(iii) f (x) = sin 2x, g (x) = 2 sin x

We know that the functions f(x) = sin 2x and g (x) = 2 sin x are periodic functions with periods π and π.

The values of these functions are tabulated below:

Values of f (x) = sin (2x) in [0, π]

x

0

π/4

π/2

3π/4

π

5π/4

3π/2

7π/4

f (x) = sin (2x)

0

1

0

-1

0

1

0

-1

0

Values of g (x) = 2 sin x in [0, π]

x

0

π/2

π

3π/2

g (x) = 2 sin x

0

1

0

-1

0

The required curve is:

(iv) f (x) = sin x/2, g (x) = sin x

We know that the functions f(x) = sin x/2 and g (x) = sin x are periodic functions with periods π and 2π.

The values of these functions are tabulated below:

Values of f (x) = sin x/2 in [0, π]

x

0

π

f (x) = sin x/2

0

1

0

-1

0

Values of g (x) = sin (x) in [0, 2π]

x

0

π/2

π

3π/2

5π/2

7π/2

g (x) = sin (x)

0

1

0

-1

0

1

0

-1

0

The required curve is:


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RD Chapter 6- Graphs of Trigonometric Functions Ex-6.1 Contributors

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