Question

the angle $\theta$ on the unit - circle (shown above), formed when the segment that connects the origin to point $p(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$ meets the positive $x$-axis, can be written as $\frac{10\pi}{a}$ radians. what is the value of $a$?

Explanation:

Step1: Identify the quadrant of the point

The point $P(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$ is in the third - quadrant.

Step2: Find the reference angle

The reference angle for a point with equal non - zero $x$ and $y$ coordinates in magnitude is $\frac{\pi}{4}$. In the third - quadrant, the angle $\theta$ from the positive $x$ - axis is $\pi+\frac{\pi}{4}=\frac{5\pi}{4}$.

Step3: Set up the equation

We are given that $\theta=\frac{10\pi}{a}$. Since $\theta = \frac{5\pi}{4}$, we set up the equation $\frac{10\pi}{a}=\frac{5\pi}{4}$.

Step4: Solve the equation for $a$

Cross - multiply: $5\pi\times a=10\pi\times4$. Divide both sides by $5\pi$: $a=\frac{40\pi}{5\pi}=8$.

Answer:

$8$