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:
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.