Sketch the graphs of the following curves on the same scale and the same axes:
(i) $y=\cos x$ and $y=\cos \left(x-\frac{\pi}{4}\right)$
(ii) $y=\cos 2 x$ and $y=\cos 2\left(x-\frac{\pi}{4}\right)$
(iii) $y=\cos x$ and $y=\cos \frac{x}{2}$
(iv) $y=\cos ^{2} x$ and $y=\cos x$
(i) First, we draw the graph of y = cos x.
Let us now draw the graph of $y=\cos \left(x-\frac{\pi}{4}\right)$.
$y=\cos \left(\mathrm{x}-\frac{\pi}{4}\right)$
$\Rightarrow y-0=\cos \left(x-\frac{\pi}{4}\right)$ ...(i)
On shifting the origin at $\left(\frac{\pi}{4}, 0\right)$, we get $:$
$\mathrm{x}=\mathrm{X}+\frac{\pi}{4}$ and $\mathrm{y}=\mathrm{Y}+0$
On subsitituting the values in $(\mathrm{i})$, we get:
$\mathrm{Y}=\cos X$
Then, we draw the graph of $Y=\cos X$ and shift it by $\frac{\pi}{4}$ to the right.
Then, we will obtain the following graph:
(ii) First, we draw the graph of y = cos 2x.
Let us now draw the graph of $y=\cos 2\left(x-\frac{\pi}{4}\right)$.
$y=\cos 2\left(\mathrm{x}-\frac{\pi}{4}\right)$
$\Rightarrow y-0=\cos 2\left(x-\frac{\pi}{4}\right)$ ...(i)
On shifting the origin at $\left(\frac{\pi}{4}, 0\right)$, we get:
$\mathrm{x}=\mathrm{X}+\frac{\pi}{4}$ and $\mathrm{y}=\mathrm{Y}+0$
On subsititut ing the values in (i), we get:
$\mathrm{Y}=\cos 2 X$
Then, we draw the graph of $Y=\cos 2 X$ and shift it by $\frac{\pi}{4}$ to the right.
Then, we will obtain the following graph:
(iii) First, we draw the graph of y = cos x.
Let us now draw the graph of $y=\cos \left(\frac{x}{2}\right)$.
$\Rightarrow y=\cos \frac{1}{2}(x)$
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:
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