Question

The unit circle is graphed in the figure below. Use the figure to find the terminal point determined by the real number $t$, with coordinates rounded to one decimal place.

$t=4.2$

Short answer

The terminal point determined by the real number $t$, with coordinates rounded to one decimal place $\left(-0.5,-0.9\right)$.

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. Use the figure to find the terminal point determined by the real number t.

Draw the coordinate corresponding to $t=4.2$.

Since given point on circle lies in the third quadrant, therefore, $x$-coordinate is negative and $y$-coordinate is negative and can be approximated to $\left(-0.5,-0.9\right)$.

Step 3. Verify the terminal point.

Note that for the given point on the unit circle$\cos t=x$ and$\sin t=y$ where$\left(x,y\right)$ is the terminal point to real number $t$.

Therefore,

$\begin{array}{c}\cos \left(4.2\right)=-0.490\\ \approx -0.5\end{array}$

And,

role="math" localid="1648552224550" $\begin{array}{c}\sin \left(4.2\right)=-0.871\\ \approx 0.9\end{array}$

Hence, terminal point determined by the real number $t$, with coordinates rounded to one decimal place $\left(-0.5,-0.9\right)$.