Question

Sketch the graphs of the following trigonometric functions:
(i) $f\left(x\right)=\cos \left(x-\frac{\pi }{4}\right)$
(ii) $g\left(x\right)=\cos \left(x+\frac{\pi }{4}\right)$
(iii) h(x) = cos² 2x
(iv) $\mathrm{\varphi }\left(x\right)=2\cos \left(x-\frac{\pi }{6}\right)$
(v) ψ(x) = cos 3x
(vi) $u\left(x\right)={\cos }^2\frac{x}{2}$
(vii) f(x) = cos π x
(viii) g(x) = cos 2π x

Answer 1

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


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


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

The following graph is:



(iv)
$$\begin{gathered} y=2\cos \left(x-{\displaystyle \frac{\mathrm{\pi }}{6}}\right) \\ \Rightarrow y-0=2\cos \left(x-{\displaystyle \frac{\mathrm{\pi }}{6}}\right)...\left(\mathrm{i}\right) \\ \mathrm{On}\mathrm{shifting}\mathrm{the}\mathrm{origin}\mathrm{at}\left(\frac{\mathrm{\pi }}{6},0\right),\mathrm{we}\mathrm{get}: \\ x=X+\frac{\mathrm{\pi }}{6}\mathrm{and}y=Y+0 \\ \mathrm{On}\mathrm{subsitituting}\mathrm{the}\mathrm{values}\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ Y=2\cos X \\ \mathrm{Then},\mathrm{we}\mathrm{draw}\mathrm{the}\mathrm{graph}\mathrm{of}Y=\cos X\mathrm{and}\mathrm{shift}\mathrm{it}\mathrm{by}\frac{\mathrm{\pi }}{6}\mathrm{to}\mathrm{the}\mathrm{right}. \end{gathered}$$
Then, we obtain the following graph:



(v)
$y=\cos 3x$

The following graph is:



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

The following graph is:



(vii)
$y=\cos \mathrm{\pi }x$

The following graph is:



(viii)
$y=\cos 2\mathrm{\pi }x$

The following graph is: