Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

In problems 13-20, $P\left(x,y\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

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

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions of $t$ for $\left(\frac{2\sqrt{2}}{3},-\frac{1}{3}\right)$ are:

$\begin{array}{c}\sin t=-\frac{1}{3}\\ \cos t=\frac{2\sqrt{2}}{3}\\ \tan t=-\frac{1}{4}\sqrt{2}\\ \csc t=-3\\ \sec t=\frac{3}{4}\sqrt{2}\\ \cot t=-2\sqrt{2}\end{array}$

Definitions.

For a point $P\left(x,y\right)$ on a  unit circle that corresponds  to $t$, the six trigonometric functions are defined as follows:  

$\begin{array}{c}\sin t=y\\ \cos t=x\\ \tan t=\frac{y}{x}\\ \csc t=\frac{1}{y}\\ \sec t=\frac{1}{x}\\ \cot t=\frac{x}{y}\end{array}$

Evaluate.

Here, $P\left(x,y\right)=P\left(\frac{2\sqrt{2}}{3},-\frac{1}{3}\right)$.

Substitute these values in the six trigonometric functions and simplify.  

Find $\sin t$.

$$\begin{gathered} \sin t =y \\ \sin t=-\frac{1}{3} \end{gathered}$$

Find $\cos t$.

$$\begin{gathered} \cos t=x \\ \cos t=\frac{2\sqrt{2}}{3} \end{gathered}$$

Find $\tan t$.

$$\begin{gathered} \tan t=\frac{y}{x} \\ \tan t=\frac{-{\displaystyle \frac{1}{3}}}{{\displaystyle \frac{2\sqrt{2}}{3}}} \\ \tan t=\frac{-{\displaystyle 1}}{{\displaystyle 2\sqrt{2}}} \\ \tan t=-\frac{1}{{\displaystyle 4}}\sqrt{2} \end{gathered}$$

Evaluate.

Find $csct$.

$$\begin{gathered} csc t=\frac{1}{y} \\ csc t=\frac{1}{-{\displaystyle \frac{1}{3}}} \\ csc t=-3 \end{gathered}$$

Find $sec t$.

$$\begin{gathered} sec t=\frac{1}{x} \\ sec t=\frac{1}{{\displaystyle \frac{2\sqrt{2}}{3}}} \\ sec t=\frac{3}{2\sqrt{2}} \\ sec t=\frac{3}{4}\sqrt{2} \end{gathered}$$

Find $cot t$.

$$\begin{gathered} cot t=\frac{x}{y} \\ cot t=\frac{{\displaystyle \frac{2\sqrt{2}}{3}}}{-{\displaystyle \frac{1}{3}}} \\ cot t=-2\sqrt{2} \end{gathered}$$