Question

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

$\left(\frac{1}{5},-\frac{2\sqrt{6}}{5}\right)$

Short answer

$\sin t=-\frac{2\sqrt{6}}{5},\cos t=\frac{1}{5}$, $\tan t=-2\sqrt{6}$.

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 $P\left(x,y\right)=\left(\frac{1}{5},-\frac{2\sqrt{6}}{5}\right)$, therefore,$x=\frac{1}{5}$ and $y=-\frac{2\sqrt{6}}{5}$.

So,

$$\begin{gathered} \sin t=y \\ =-\frac{2\sqrt{6}}{5} \end{gathered}$$

And,

$$\begin{gathered} \cos t=x \\ =\frac{1}{5} \end{gathered}$$

Step 3. Find tant.

To find$\tan t$ substitute$\frac{1}{5}$ for $x$ and$-\frac{2\sqrt{6}}{5}$ for $y$ into the expression$\tan t=\frac{y}{x}$ and simplify.

$$\begin{gathered} \tan t=\frac{y}{x} \\ =\frac{\left(-\frac{2\sqrt{6}}{5}\right)}{\left(\frac{1}{5}\right)} \\ =-2\sqrt{6} \end{gathered}$$

Hence, $\tan t=-2\sqrt{6}$.