Question

Sketch the graphs of the following curves on the same scale and the same axes:
(i) $y=\cos x\mathrm{and}y=\cos \left(x-\frac{\pi }{4}\right)$
(ii) $y=\cos 2x\mathrm{and}y=\cos 2\left(x-\frac{\pi }{4}\right)$
(iii) $y=\cos x\mathrm{and}y=\cos \frac{x}{2}$
(iv) $y={\cos }^{2}x\mathrm{and}y=\cos x$

Answer 1

(i)
First, we draw the graph of y = cos x.
Let us now draw the graph of $y=\cos \left(x-\frac{\mathrm{\pi }}{4}\right)$.
$$\begin{gathered} y=\cos \left(\mathrm{x}-{\displaystyle \frac{\mathrm{\pi }}{4}}\right) \\ \Rightarrow y-0=\cos \left(x-\frac{\mathrm{\pi }}{4}\right)...\left(\mathrm{i}\right) \\ On\mathrm{shifting}\mathrm{the}\mathrm{origin}\mathrm{at}\left(\frac{\mathrm{\pi }}{4},0\right),\mathrm{we}get: \\ \mathrm{x}=\mathrm{X}+\frac{\mathrm{\pi }}{4}\mathrm{and}\mathrm{y}=\mathrm{Y}+0 \\ On\mathrm{subsititut}ing\mathrm{the}\mathrm{values}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ \mathrm{Y}=\cos X \\ \mathrm{Then},\mathrm{we}\mathrm{draw}\mathrm{the}\mathrm{graph}\mathrm{of}Y=\cos X\mathrm{and}\mathrm{shift}\mathrm{it}\mathrm{by}\frac{\mathrm{\pi }}{4}\mathrm{to}\mathrm{the}\mathrm{right}. \end{gathered}$$
Then, we will obtain the following graph:



(ii)
First, we draw the graph of y = cos 2x.
Let us now draw the graph of $y=\cos 2\left(x-\frac{\mathrm{\pi }}{4}\right)$.
$$\begin{gathered} y=\cos 2\left(\mathrm{x}-{\displaystyle \frac{\mathrm{\pi }}{4}}\right) \\ \Rightarrow y-0=\cos 2\left(x-\frac{\mathrm{\pi }}{4}\right)...\left(\mathrm{i}\right) \\ On\mathrm{shifting}\mathrm{the}\mathrm{origin}\mathrm{at}\left(\frac{\mathrm{\pi }}{4},0\right),\mathrm{we}get: \\ \mathrm{x}=\mathrm{X}+\frac{\mathrm{\pi }}{4}\mathrm{and}\mathrm{y}=\mathrm{Y}+0 \\ On\mathrm{subsititut}ing\mathrm{the}\mathrm{values}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ \mathrm{Y}=\cos 2X \\ \mathrm{Then},\mathrm{we}\mathrm{draw}\mathrm{the}\mathrm{graph}\mathrm{of}Y=\cos 2X\mathrm{and}\mathrm{shift}\mathrm{it}\mathrm{by}\frac{\mathrm{\pi }}{4}\mathrm{to}\mathrm{the}\mathrm{right}. \end{gathered}$$
Then, we will obtain the following graph:



(iii)

First, we draw the graph of y = cos x.
Let us now draw the graph of $y=\cos \left(\frac{x}{2}\right)$.
$\Rightarrow y=\cos \frac{1}{2}\left(x\right)$
Then, we will obtain the following graph:


(iv)
First, we draw the graph of y = cos² x.
Let us now draw the graph of y = cos x.

Then, we will obtain the following graph: