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Class 11 RD Sharma Solutions – Chapter 6 Graphs of Trigonometric Functions – Exercise 6.2

Last Updated : 08 May, 2021

Question 1: Sketch the following graphs:

(i) y = cos (x+ $\frac{\pi}{4}$ )

Solution:

To obtain this graph y-0 = cos (x+ $\frac{\pi}{4}$ ),
Shifting the origin at $(-\frac{\pi}{4},0)$ , we have
X = x+ $\frac{\pi}{4}$ and Y = y-0
Substituting these values, we get
Y = cos X
First we draw the graph of Y = cos X and shift it by $\frac{\pi}{4}$ to the left.

(ii) y = cos (x- $\frac{\pi}{4}$ )

Solution:

To obtain this graph y-0 = cos (x- $\frac{\pi}{4}$ ),
Shifting the origin at $(\frac{\pi}{4},0)$ , we have
X = $x-\frac{\pi}{4}$ and Y = y-0
Substituting these values, we get
Y = cos X
First we draw the graph of Y = cos X and shift it by $\frac{\pi}{4}$ to the right.

(iii) y = $3 \hspace{0.1cm}cos (2x-1)$

Solution:

To obtain this graph y-0 = 3 cos 2(x- $\frac{1}{2}$ ),
Shifting the origin at $(\frac{1}{2},0)$ , we have
X = $x-\frac{1}{2}$ and Y = y-0
Substituting these values, we get
Y = 3 cos 2X
First we draw the graph of Y = 3 cos 2X and shift it by $\frac{1}{2}$ to the right.
The maximum and minimum values of y are 3 and -3 respectively.

(iv) $y = 2 \hspace{0.1cm}cos (x-\frac{\pi}{2})$

Solution:

To obtain this graph y-0 = $2 \hspace{0.1cm}cos (x-\frac{\pi}{2})$ ,
Shifting the origin at $(\frac{\pi}{2},0)$ , we have
X = $x-\frac{\pi}{2}$ and Y = y-0
Substituting these values, we get
Y = 2 cos X
First we draw the graph of Y = 2 cos X and shift it by $\frac{\pi}{2}$ to the right.
The maximum and minimum values of y are 2 and -2 respectively.

Question 2: Sketch the graphs of the following functions on the same scale:

(i) y = cos x, y = cos $(x-\frac{\pi}{4})$

Solution:

Graph 1:
y = cos x
Graph 2:
To obtain this graph y-0 = cos $(x-\frac{\pi}{4})$ ,
Shifting the origin at $(\frac{\pi}{4},0)$ , we have
X = $x-\frac{\pi}{4}$ and Y = y-0
Substituting these values, we get
Y = cos X
First we draw the graph of Y = cos X and shift it by $\frac{\pi}{4}$ to the right.
The graph y = cos x and $y = cos (x-\frac{\pi}{4})$ are on same axes are as follows:

(ii) y = cos 2x, y = cos $2(x-\frac{\pi}{4})$

Solution:

Graph 1:
To obtain this graph y = cos 2x,
First we draw the graph of y = cos x and then divide the x-coordinates of the points where it crosses x-axis by 2.
Graph 2:
To obtain this graph y-0 = cos $2(x-\frac{\pi}{4})$ ,
Shifting the origin at $(\frac{\pi}{4},0)$ , we have
X = $x-\frac{\pi}{4}$ and Y = y-0
Substituting these values, we get
Y = cos 2X
First we draw the graph of Y = cos 2X and shift it by $\frac{\pi}{4}$ to the right.
The graph y = cos 2x and y = cos $2(x-\frac{\pi}{4})$ are on same axes are as follows:

(iii) y = cos x, $y = cos (\frac{x}{2})$

Solution:

Graph 1:
y = cos x
Graph 2:
To obtain this graph y = $cos (\frac{x}{2})$ ,
First we draw the graph of y = cos x and then multiply the x-coordinates of the points where it crosses x-axis by 2.
The graph y = cos x and y = $cos (\frac{x}{2})$ are on same axes are as follows: