91Ó°ÊÓ

The given functions are $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]\text{and}y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$.

A function of the form:

$y=a\sin b\left(\mathrm{θ}-h\right)+k,y=a\cos b\left(\mathrm{θ}-h\right)+k$has vertical shift $\left(k\right)$. And, a period of $\frac{360°}{\left|b\right|}\text{or}\frac{2\mathrm{π}}{\left|b\right|}$ for sine, cosecant, secant and cosine functions and a period of $\frac{180°}{\left|b\right|}\text{or}\frac{\mathrm{π}}{\left|b\right|}$ for tangent and cotangent function. The phase shift for the functions is $\left(h\right)$.

The amplitude of secant, cosecant, tangent and cotangent functions is not defined.

The equation for the midline is written as, $y=k$. A midline is a new reference line when the parent graph is stretched vertically up or down and then the graph oscillates about new reference line called the midline.

The given functions are $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]\text{and}y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$.

The parent function $y=\sin \left(\mathrm{θ}\right)$ is shifted towards right by a phase shift of $\frac{\mathrm{π}}{2}$ to get the function $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]$.

The parent function $y=\cos \left(\mathrm{θ}\right)$ is shifted towards left by a phase shift of $-\frac{3\mathrm{π}}{2}$ to get the function $y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$.

It can also be seen from the graph drawn of the functions $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]\text{and}y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$.

The graph of the functions $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]\text{and}y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$ are shown below.

From the graph of the function $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]\text{and}y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$ it can be interpreted that the graph of the functions $y=-\sin \left[\frac{1}{4}\left(\mathrm{θ}-\frac{\mathrm{π}}{2}\right)\right]\text{and}y=\cos \left[\frac{1}{4}\left(\mathrm{θ}+\frac{3\mathrm{π}}{2}\right)\right]$ overlap each other.