Answer
a) The value is $0$.
b) The value is $0$.
Work Step by Step
(a)
The point corresponding to $t=\frac{\pi }{2}$ has the coordinates $\left( 1,0 \right)$
Thus, use $x=1$ and $y=0$
Such that,
$\cos \left( \frac{\pi }{2} \right)=y=0$
Therefore, the value of the trigonometric function $\cos \left( \frac{\pi }{2} \right)$ is $0$.
(b)
We know that the periodic properties of sine and cosine functions are,
$\sin \left( t+2\pi \right)=\sin \left( t \right)$ and $\text{cos}\left( t+2\pi \right)=\cos \left( t \right)$.
So,
$\begin{align}
& \cos \left( \frac{9\pi }{2} \right)=\cos \left( \frac{\pi }{2}+4\pi \right) \\
& =\cos \left( \frac{\pi }{2}+2\left( 2\pi \right) \right) \\
& =\cos \frac{\pi }{2}
\end{align}$
Now, put $\cos \left( \frac{\pi }{2} \right)=0$.
So,
$\cos \left( \frac{9\pi }{2} \right)=0$
Thus, the value of the trigonometric function $\cos \left( \frac{9\pi }{2} \right)$ is $0$.
