Exercise 6.2 of Chapter 6, discusses problems based on the graph of a cosine function. The exercise problems are solved by an experienced faculty having in-depth knowledge about these concepts. In addition, the solutions are designed as per the weightage allotted in the board exam. While solving problems students can use the solutions pdf for a quick reference and guidance, which helps in scoring good marks in their exams. RD Sharma Class 11 Solutions are available in the pdf format which can be downloaded easily from the links given below.

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**1. Sketch the graphs of the following trigonometric functions:**

**(i) f (x) = cos (x â€“ Ï€/4)**

**(ii) g (x) = cosÂ (x + Ï€/4)**

**(iii) h (x) = cos ^{2}Â 2x**

**(iv) Ï• (x) = 2 cos (x â€“ Ï€/6)**

**(v) Ïˆ (x) = cos (3x)**

**(vi) u (x) = cos ^{2} x/2**

**(vii) f (x) = cos Ï€x**

**(viii) g (x) = cos 2Ï€ x**

**Solution:**

**(i) **f (x) = cos (x â€“ Ï€/4)

We know that g (x) = cos x is a periodic function with period 2Ï€.

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

x |
0 (A) |
Ï€/4 (B) |
Ï€/2 (C) |
3Ï€/4 (D) |
Ï€ (E) |
5Ï€/4 (F) |
3Ï€/2 (G) |
7Ï€/4 (H) |

f (x) = cos (x â€“ Ï€/4) |
1/âˆš2 = 0.7 |
1 |
1/âˆš2 = 0.7 |
0 |
-1/âˆš2 = -0.7 |
-1 |
-1/âˆš2 = -0.7 |
0 |

The required curve is:

**(ii)** g (x) = cosÂ (x + Ï€/4)

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

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

x |
0 (A) |
Ï€/4 (B) |
Ï€/2 (C) |
3Ï€/4 (D) |
Ï€ (E) |
5Ï€/4 (F) |
3Ï€/2 (G) |
7Ï€/4 (H) |

g (x) = cos (x + Ï€/4) |
1/âˆš2 = 0.7 |
0 |
-1/âˆš2 = -0.7 |
-1 |
-1/âˆš2 = -0.7 |
0 |
1/âˆš2 = 0.7 |
1 |

The required curve is:

**(iii)** h (x) = cos^{2}Â 2x

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

So, h (x) = cos^{2}Â 2x is a periodic function with period Ï€. So, we will draw the graph of h (x) = cos^{2}Â 2x in the interval [0, Ï€]. The values of h (x) = cos^{2}Â 2x at various points in [0, Ï€] are listed in the following table:

x |
0 (A) |
Ï€/4 (B) |
Ï€/2 (C) |
3Ï€/4 (D) |
Ï€ (E) |
5Ï€/4 (F) |
3Ï€/2 (G) |

h (x) = cos |
1 |
0 |
1 |
0 |
1 |
0 |
1 |

The required curve is:

**(iv)** Ï• (x) = 2 cos (x â€“ Ï€/6)

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

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

x |
0 (A) |
Ï€/3 (B) |
2Ï€/3 (C) |
Ï€ (D) |
4Ï€/3 (E) |
5Ï€/3 (F) |

Ï• (x) = 2 cos (x â€“ Ï€/6) |
âˆš3 = 1.73 |
âˆš3 = 1.73 |
0 |
-âˆš3 = -1.73 |
-âˆš3 = -1.73 |
0 |

The required curve is:

**(v)** Ïˆ (x) = cos (3x)

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

So, Ïˆ (x) = cos (3x) is a periodic function with period 2Ï€/3. So, we will draw the graph of Ïˆ (x) = cos (3x) in the interval [0, 2Ï€/3]. The values of Ïˆ (x) = cos (3x) at various points in [0, 2Ï€/3] are listed in the following table:

x |
0 (A) |
Ï€/6 (B) |
Ï€/3 (C) |
Ï€/2 (D) |
2Ï€/3 (E) |
5Ï€/6 (F) |

Ïˆ (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 with period 2Ï€.

So, u (x) = cos^{2}Â (x/2) is a periodic function with period Ï€. So, we will draw the graph of u (x) = cos^{2}Â (x/2) in the interval [0, Ï€]. The values of u (x) = cos^{2}Â (x/2) at various points in [0, Ï€] are listed in the following table:

x |
0 (A) |
Ï€ (B) |
2Ï€ (C) |
3Ï€ (D) |

u (x) = cos |
1 |
0 |
1 |
0 |

The required curve is:

**(vii) **f (x) = cos Ï€x

We know that g (x) = cos x is a periodic function with period 2Ï€.

So, f (x) = cos (Ï€x) is a periodic function with period 2. So, we will draw the graph of f (x) = cos (Ï€x) in the interval [0, 2]. The values of f (x) = cos (Ï€x) 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) |
5/2 (F) |

f (x) = cos Ï€x |
1 |
0 |
-1 |
0 |
1 |
0 |

The required curve is:

**(viii)** g (x) = cos 2Ï€ x

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

So, g (x) = cos (2Ï€x) is a periodic function with period 1. So, we will draw the graph of g (x) = cos (2Ï€x) in the interval [0, 1]. The values of g (x) = cos (2Ï€x) at various points in [0, 1] are listed in the 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 2Ï€ x |
1 |
0 |
-1 |
0 |
1 |
0 |
-1 |
0 |
1 |

The required curve is:

**2. Sketch the graphs of the following curves on the same scale and the same axes:**

**(i) y = cos x and y = cos (x â€“ Ï€/4)Â **

**(ii) y = cos 2x and y = cos (x – **Ï€/4**)Â **

**(iii) y = cos x and y = cos x/2Â **

**(iv) y = cos ^{2}Â x and y = cos x**

**Solution:**

**(i) **y = cos x and y = cos (x â€“ Ï€/4)Â

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

The values of these functions are tabulated below:

x |
0 |
Ï€/4 |
Ï€/2 |
3Ï€/4 |
Ï€ |
5Ï€/4 |
3Ï€/2 |
7Ï€/4 |

y = cos x |
1 |
1/âˆš2 = 0.7 |
0 |
-1/âˆš2 = -0.7 |
-1 |
-1/âˆš2 = -0.7 |
0 |
1 |

y = cos (x â€“ Ï€/4)Â |
1/âˆš2 = 0.7 |
1 |
1/âˆš2 = 0.7 |
0 |
-1/âˆš2 = -0.7 |
-1 |
-1/âˆš2 = -0.7 |
0 |

The required curve is:

**(ii)** y = cos 2x and y = cos 2(x – Ï€/4)**Â **

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

The values of these functions are tabulated below:

x |
0 |
Ï€/4 |
Ï€/2 |
3Ï€/4 |
Ï€ |
5Ï€/4 |
3Ï€/2 |
7Ï€/4 |

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

y = cos 2 (x â€“ Ï€/4)Â |
0 |
1 |
0 |
-1 |
0 |
1 |
0 |
-1 |

The required curve is:

**(iii)** y = cos x and y = cos x/2**Â **

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

The values of these functions are tabulated below:

x |
0 |
Ï€/2 |
Ï€ |
3Ï€/2 |
2Ï€ |

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

y = cos x/2 |
1 |
1/âˆš2 = 0.7 |
0 |
-1/âˆš2 = -0.7 |
-1 |

The required curve is:

**(iv)** y = cos^{2}Â x and y = cos x

We know that the functions y = cos^{2}Â x and y = cos x are periodic functions with period 2Ï€.

The values of these functions are tabulated below:

x |
0 |
Ï€/2 |
Ï€ |
3Ï€/2 |
2Ï€ |

y = cos |
1 |
0 |
1 |
0 |
1 |

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

The required curve is: