Question

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

$t=-3\pi$

Short answer

The terminal point on the unit circle is determined by$t=-3\pi$ is$P\left(x,y\right)=\left(-1,0\right)$.

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 the clockwise direction and if $t<0$, then$\left|t\right|$ is the distance along the unit circle in the 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=−π.

The circumference of the unit circle is $C=2\pi$. So, if a point starts at$\left(1,0\right)$and moves clockwise halfway around the unit circle $\frac{1}{2}\left(-2\pi \right)=-\pi$, the point will come to localid="1648241444123" $\left(-1,0\right)$.

Step 3. Simplify and state the conclusion.

Therefore, to move thrice halfway around the circle in the clockwise direction, it travels a distance of $3\left(-\pi \right)=-3\pi$, it will travel to the same point as when $t=-\pi$.

So, the terminal point is determined by$-3\pi$is localid="1648241463528" $\left(-1,0\right)$.