Sketch the graphs of the following trigonometric functions:

Question:

Sketch the graphs of the following trigonometric functions:

(i) $f(x)=\cos \left(x-\frac{\pi}{4}\right)$

(ii) $g(x)=\cos \left(x+\frac{\pi}{4}\right)$

(iii) $h(x)=\cos ^{2} 2 x$

(iv) $\phi(x)=2 \cos \left(x-\frac{\pi}{6}\right)$

(v) $\Psi(x)=\cos 3 x$

(vi) $u(x)=\cos ^{2} \frac{x}{2}$

(vii) $f(x)=\cos \pi x$

(viii) $g(x)=\cos 2 \pi x$

Solution:

(i) $y=\cos \left(\mathrm{x}-\frac{\pi}{4}\right)$

$\Rightarrow y-0=\cos \left(x-\frac{\pi}{4}\right)$             ...(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 subsititut $i n g$ the values in (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 obtain the following graph:

(ii) $y=\cos \left(\mathrm{x}+\frac{\pi}{4}\right)$

$\Rightarrow y-0=\cos \left(x+\frac{\pi}{4}\right)$              ...(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 left.

Then, we obtain the following graph:

(iii) $y=\cos ^{2} 2 x$

The following graph is:

(iv) $y=2 \cos \left(x-\frac{\pi}{6}\right)$

$\Rightarrow y-0=2 \cos \left(x-\frac{\pi}{6}\right)$                ..(i)

On shifting the origin at $\left(\frac{\pi}{6}, 0\right)$, we get:

$x=X+\frac{\pi}{6}$ and $y=Y+0$

On subsitituting the values in $(\mathrm{i})$, we get:

$Y=2 \cos X$

Then, we draw the graph of $Y=\cos X$ and shift it by $\frac{\pi}{6}$ to the right.

Then, we obtain the following graph:

(v) $y=\cos 3 x$

The following graph is:

(vi) $y=\cos ^{2} \frac{x}{2}$

The following graph is:

(vii) $y=\cos \pi x$

The following graph is:

(viii)$y=\cos 2 \pi x$

The following graph is:

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