Question

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

$\left(-\frac{3}{5},-\frac{4}{5}\right)$

Short answer

$\sin t=-\frac{4}{5},\cos t=-\frac{3}{5}$, $\tan t=\frac{4}{3}$.

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 $P\left(x,y\right)=\left(-\frac{3}{5},-\frac{4}{5}\right)$, therefore,$x=-\frac{3}{5}$ and $y=-\frac{4}{5}$.

So,

role="math" localid="1648683199471" $$\begin{gathered} \sin t=y \\ =-\frac{4}{5} \end{gathered}$$

And,

role="math" localid="1648683241131" $$\begin{gathered} \cos t=x \\ =-\frac{3}{5} \end{gathered}$$

Step 3. Find tant.

To find $\tan t$ substitute $-\frac{3}{5}$for $x$and $-\frac{4}{5}$for $y$into the expression $\tan t=\frac{y}{x}$and simplify.

$$\begin{gathered} \tan t=\frac{y}{x} \\ =\frac{\left(-\frac{4}{5}\right)}{\left(-\frac{3}{5}\right)} \\ =\frac{4}{3} \end{gathered}$$

Hence, $\tan t=\frac{4}{3}$.