Answer
The way to use the unit circle to find the value of the trigonometric function at $\frac{\pi }{4}$ is to find the points on the unit circle that corresponds to $t=\frac{\pi }{4}$.
Work Step by Step
First, find the points on the unit circle $\left( a,b \right)$ that corresponds to$t=\frac{\pi }{4}$.
Consider the unit circle as shown in the figure:
The point $P$ lies on the line $y=x$
Consider the standard equation of the circle${{x}^{2}}+{{y}^{2}}=1$ .
Since,
$y=x$
Substitute $y$ for $x$ in above equation to get:
$\begin{align}
& {{y}^{2}}+{{y}^{2}}=1 \\
& 2{{y}^{2}}=1 \\
& y=\frac{1}{\sqrt{2}} \\
& y=\frac{\sqrt{2}}{2}
\end{align}$
Thus, the point is $\left( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2} \right)$
Substitute these values in the formula of the trigonometric functions to get the required values.
