Question

The terminal point$P\left(x,y\right)$determined by a real number $t$ is given. Find$\sin t,\cos t$and$\tan t$.

$\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$

Short answer

$\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2}$, $\tan t=-\sqrt{3}$.

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. Find sint and cost.

Note that for the given point $t$ on the unit circle $\cos t=x$,$\sin t=y$ and$\tan t=\frac{y}{x}$ where$\left(x,y\right)$ is the terminal point to real number $t$. Since $P\left(x,y\right)=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$, therefore,$x=-\frac{1}{2}$ and $y=\frac{\sqrt{3}}{2}$.

So,

role="math" localid="1648684187247" $$\begin{gathered} \sin t=y \\ =\frac{\sqrt{3}}{2} \end{gathered}$$

And,

role="math" localid="1648684220750" $$\begin{gathered} \cos t=x \\ =-\frac{1}{2} \end{gathered}$$

Step 3. Find tant.

To find $\tan t$ substitute $-\frac{1}{2}$for $x$and $\frac{\sqrt{3}}{2}$for $y$into the expression $\tan t=\frac{y}{x}$and simplify.

$$\begin{gathered} \tan t=\frac{y}{x} \\ =\frac{\left(\frac{\sqrt{3}}{2}\right)}{\left(-\frac{1}{2}\right)} \\ =-\sqrt{3} \end{gathered}$$

Hence, $\tan t=-\sqrt{3}$.