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

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: