Question

Find the terminal point $P\left(x,y\right)$ on the unit circle determined by the given value of $t$.

$t=\frac{3\pi }{2}$

Short answer

The terminal point on the unit circle determinedby$t=\frac{3\pi }{2}$ is$P\left(x,y\right)=P\left(0,-1\right)$.

Step-by-step solution

Step 1. State the definition of the terminal point on the unit circle.

Starting at the point $\left(1,0\right)$, if $t\ge 0$, then $t$ is the distance along the unit circle in clockwise direction and if $t<0$, then$\left|t\right|$ is the distance along the unit circle in clockwise direction. Here, $t$ is a real number and this distance generates a point$P\left(x,y\right)$ on the unit circle. The point$P\left(x,y\right)$ obtained in this way is called the terminal point determined by the real number $t$.

Step 2. State interpretation of terminal point when t=π2.

The circumference of the unit circle is $C=2\pi$. So, if a point starts at $\left(1,0\right)$and moves a quarter of the distance around the circle counterclockwise then it travels to the pointlocalid="1648242562103" $\left(0,1\right)$i.e., it travels a distance of $\frac{1}{4}\left(2\pi \right)=\frac{\pi }{2}$.

Step 3. Simplify and state the conclusion.

Therefore, if the point moves quarterly thrice around the circle, then it travels a distance of $3\left(\frac{\pi }{2}\right)=\frac{3}{2}\pi$.

So, the terminal point determined by$\frac{3\pi }{2}$ is role="math" localid="1648242172439" $P\left(0,-1\right)$, as depicted below.