RD Chapter 6- Graphs of Trigonometric Functions Ex-6.2 |
RD Chapter 6- Graphs of Trigonometric Functions Ex-6.3 |

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 |
| 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 | 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) = sin^{2} 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) = sin^{2} x is a periodic function withperiod 2π. So, we will draw the graph of u (x) = sin^{2} xin the interval [0, 2π]. The values of u (x) = sin^{2} 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) = sin | 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:

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 | 2π |

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 | 2π |

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 | 2π |

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 | 2π |

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 | 2π |

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 | 2π | 3π | 4π |

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 | 2π | 5π/2 | 3π | 7π/2 | 4π |

g (x) = sin (x) | 0 | 1 | 0 | -1 | 0 | 1 | 0 | -1 | 0 |

The required curve is:

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