Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

First we are going to convert radians into degrees. We know that we will get degrees when we multiply radians by $\dfrac{180\text{\textdegree}}{\pi}$.

$$\frac{3\pi}{8}\cdot \frac{180\text{\textdegree}}{\pi}=67.5\text{\textdegree}$$

We are going to use half-angle formula for $\sin$ which is

$$\sin{\frac{A}{2}}=\pm\sqrt{\frac{1-\cos{A}}{2}}$$

Now we need to find the value of $A$.

We are going to rewrite $\sin{67.5\text{\textdegree}}$ as $\sin{\dfrac{A}{2}}$.

$$\sin{67.5\text{\textdegree}}=\sin{\frac{135\text{\textdegree}}{2}}$$

We can see that $A=135\text{\textdegree}$.

We are going to determine if is square root positive or negative. Since $135\text{\textdegree}$ is in the Second Quadrant and we know that all $\sin$ values are positive in Second Quadrant, we will have positive square root.

Now we are going to substitute $135\text{\textdegree}$ for $A$ into expression.

$$\begin{align*} \sin{67.5\text{\textdegree}}&=\sqrt{\frac{1-\cos{135\text{\textdegree}}}{2}} && \text{Substitute.} \end{align*}$$

We are going to find the value of $\cos{135\text{\textdegree}}$ on the Unit Circle.

The terminal side intersect the Unit Circle in the point $(\frac{-\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. Where on the Unit Circle $x=\cos{\theta}$ and $y=\sin{\theta}$.

This indicates that $\cos{135\text{\textdegree}}=-\frac{\sqrt{2}}{2}$.