RD Chapter 6- Graphs of Trigonometric Functions Ex-6.1 |
RD Chapter 6- Graphs of Trigonometric Functions Ex-6.3 |

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) = cos^{2} 2x

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

So, h (x) = cos^{2} 2x is a periodicfunction with period π. So, we will draw the graph of h (x) = cos^{2} 2xin the interval [0, π]. The values of h (x) = cos^{2} 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) = cos | 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) = cos^{2} x/2

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

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

x | 0 (A) | π (B) | 2π (C) | 3π (D) |

u (x) = cos | 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 = cos^{2} x and y = cos x

We know that the functions y = cos^{2} 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 = cos | 1 | 0 | 1 | 0 | 1 |

y = cos x | 1 | 0 | -1 | 0 | 1 |

The required curve is:

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