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RD Chapter 6- Graphs of Trigonometric Functions Ex-6.2 Interview Questions Answers

Question 1 :
Sketch the graphs of the following trigonometric functions:
(i) f (x) = cos (x – π/4)
(ii) g (x) = cos (x + π/4)
(iii) h (x) = cos2 2x
(iv) ϕ (x) = 2 cos (x – π/6)
(v) ψ (x) = cos (3x)
(vi) u (x) = cos2 x/2
(vii) f (x) = cos πx
(viii) g (x) = cos 2π x

Answer 1 :

(i) f (x) = cos (x – π/4)
We know that g (x) = cos x is a periodic function with period 2π.
So, f (x) = cos (x – π/4) is a periodic function with period π. So, we will draw the graph of f (x) = cos (x – π/4) in the interval [0, π]. The values of f (x) = cos (x – π/4) at various points in [0, π] are listed in the following table:

x

0 (A)

π/4 (B)

π/2 (C)

3π/4 (D)

π (E)

5π/4 (F)

3π/2 (G)

7π/4 (H)

f (x) = cos (x – π/4)

1/√2 = 0.7

1

1/√2 = 0.7

0

-1/√2 = -0.7

-1

-1/√2 = -0.7

0

The required curve is:

(ii) g (x) = cos (x + π/4)

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

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

x

0 (A)

π/4 (B)

π/2 (C)

3π/4 (D)

π (E)

5π/4 (F)

3π/2 (G)

7π/4 (H)

g (x) = cos (x + π/4)

1/√2 = 0.7

0

-1/√2 = -0.7

-1

-1/√2 = -0.7

0

1/√2 = 0.7

1

The required curve is:

(iii) h (x) = cos2 2x

We know that f (x) = cos x is a periodic function withperiod 2π.

So, h (x) = cos2 2x is a periodicfunction with period π. So, we will draw the graph of h (x) = cos2 2xin the interval [0, π]. The values of h (x) = cos2 2x atvarious points in [0, π] are listed in the following table:

x

0 (A)

π/4 (B)

π/2 (C)

3π/4 (D)

π (E)

5π/4 (F)

3π/2 (G)

h (x) = cos2 2x

1

0

1

0

1

0

1

The required curve is:

(iv) ϕ (x) = 2 cos (x – π/6)

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

So, ϕ (x) = 2cos (x – π/6) is a periodic function with period π. So, we will draw the graph of ϕ (x) = 2cos (x – π/6) in the interval [0, π]. The values of ϕ (x) = 2cos (x – π/6) at various points in [0, π] are listed in the following table:

x

0 (A)

π/3 (B)

2π/3 (C)

π (D)

4π/3 (E)

5π/3 (F)

ϕ (x) = 2 cos (x – π/6)

√3 = 1.73

√3 = 1.73

0

-√3 = -1.73

-√3 = -1.73

0

The required curve is:

(v) ψ (x) = cos (3x)

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

So, ψ (x) = cos (3x) is a periodic function with period 2π/3. So, we will draw the graph of ψ (x) = cos (3x) in the interval [0, 2π/3]. The values of ψ (x) = cos (3x) at various points 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)

5π/6 (F)

ψ (x) = cos (3x)

1

0

-1

0

1

0

The required curve is:

(vi) u (x) = cos2 x/2

We know that f (x) = cos x is a periodic function withperiod 2π.

So, u (x) = cos2 (x/2) is a periodicfunction with period π. So, we will draw the graph of u (x) = cos2 (x/2)in the interval [0, π]. The values of u (x) = cos2 (x/2) atvarious points in [0, π] are listed in the following table:

x

0 (A)

π (B)

2π (C)

3π (D)

u (x) = cos2 x/2

1

0

1

0

The required curve is:

(vii) f (x) = cos πx

We know that g (x) = cos x is a periodic function withperiod 2π.

So, f (x) = cos (πx) is a periodic function withperiod 2. So, we will draw the graph of f (x) = cos (πx) in the interval [0,2]. The values of f (x) = cos (πx) at various points in [0, 2] are listed inthe following table:

x

0 (A)

1/2 (B)

1 (C)

3/2 (D)

2 (E)

5/2 (F)

f (x) = cos πx

1

0

-1

0

1

0

The required curve is:

(viii) g (x) = cos 2π x

We know that f (x) = cos x is a periodic function withperiod 2π.

So, g (x) = cos (2πx) is a periodic function withperiod 1. So, we will draw the graph of g (x) = cos (2πx) in the interval [0,1]. The values of g (x) = cos (2πx) at various points in [0, 1] are listed inthe following table:

x

0 (A)

1/4 (B)

1/2 (C)

3/4 (D)

1 (E)

5/4 (F)

3/2 (G)

7/4 (H)

2

g (x) = cos 2π x

1

0

-1

0

1

0

-1

0

1

The required curve is:

Question 2 :
Sketch the graphs of the following curves on the same scale and the same axes:
(i) y = cos x and y = cos (x – π/4) 
(ii) y = cos 2x and y = cos (x – π/4) 
(iii) y = cos x and y = cos x/2 
(iv) y = cos2 x and y = cos x

Answer 2 :

(i) y = cos x and y = cos (x – π/4) 
We know that the functions y = cos x and y = cos (x – π/4) are periodic functions with periods π and π.
The values of these functions are tabulated below:

x

0

π/4

π/2

3π/4

π

5π/4

3π/2

7π/4

y = cos x

1

1/√2 = 0.7

0

-1/√2 = -0.7

-1

-1/√2 = -0.7

0

1

y = cos (x – π/4) 

1/√2 = 0.7

1

1/√2 = 0.7

0

-1/√2 = -0.7

-1

-1/√2 = -0.7

0

The required curve is:

(ii) y = cos 2x and y = cos 2(x – π/4) 

We know that the functions y = cos 2x and y = cos 2(x – π/4) are periodic functions with periods π and π.

The values of these functions are tabulated below:

x

0

π/4

π/2

3π/4

π

5π/4

3π/2

7π/4

y = cos x

1

0

-1

0

1

0

-1

0

y = cos 2 (x – π/4) 

0

1

0

-1

0

1

0

-1

The required curve is:

(iii) y = cos x and y = cos x/2 

We know that the functions y = cos x and y = cos (x/2) are periodic functions with periods π and π.

The values of these functions are tabulated below:

x

0

π/2

π

3π/2

y = cos x

1

0

-1

0

1

y = cos x/2

1

1/√2 = 0.7

0

-1/√2 = -0.7

-1

The required curve is:

(iv) y = cos2 x and y = cos x

We know that the functions y = cos2 xand y = cos x are periodic functions with period 2π.

The values of these functions are tabulated below:

x

0

π/2

π

3π/2

y = cos2 x

1

0

1

0

1

y = cos x

1

0

-1

0

1

The required curve is:


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RD Chapter 6- Graphs of Trigonometric Functions Ex-6.2 Contributors

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