Question

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

$t=4\pi$

Short answer

The terminal point on the unit circle is determinedby$t=4\pi$ 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=2π.

The circumference of the unit circle is $C=2\pi$. So, if a point starts at$\left(1,0\right)$ and moves counterclockwise all the way around the unit circle and returns to$\left(1,0\right)$ it travels a distance of $2\pi$.

Step 3. State the conclusion.

Therefore, to move twice around the circle, the point travels a distance of $2\left(2\pi \right)=4\pi$.

So, the terminal point is determined by$4\pi$ is $\left(1,0\right)$.