RD Chapter 6 Graphs of Trigonometric Functions Ex 6.2 Solutions
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:
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 | 2π |
| 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 | 2π |
| y = cos2 x | 1 | 0 | 1 | 0 | 1 |
| y = cos x | 1 | 0 | -1 | 0 | 1 |
The required curve is:
