Question

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

Solution

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Step 1
1 of 3

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

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