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 $t$.

$\left(-\frac{2}{5},\frac{\sqrt{21}}{5}\right)$.

Short answer

The exact values of the six trigonometric functions of $t$for $\left(-\frac{2}{5},\frac{\sqrt{21}}{5}\right)$are:

$\begin{array}{c}\sin t=\frac{\sqrt{21}}{5}\\ \cos t=-\frac{2}{5}\\ \tan t=-\frac{\sqrt{21}}{2}\\ \csc t=\frac{5}{21}\sqrt{21}\\ \sec t=-\frac{5}{2}\\ \cot t=-\frac{2}{21}\sqrt{21}\end{array}$

Step-by-step solution

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, $P\left(x,y\right)=P\left(-\frac{2}{5},\frac{\sqrt{21}}{5}\right)$.

Substitute these values in the six trigonometric functions and simplify.

Find $\sin t$.

$\begin{array}{rcl}\sin t& =& y\\ \sin t& =& \frac{\sqrt{21}}{5}\end{array}$

Find $\cos t$.

$\begin{array}{rcl}\cos t& =& x\\ \cos t& =& -\frac{2}{5}\end{array}$

Find $\tan t$.

$$\begin{gathered} \tan t=\frac{y}{x} \\ \tan t=\frac{{\displaystyle \frac{\sqrt{21}}{5}}}{-{\displaystyle \frac{2}{5}}} \\ \tan t=-\frac{\sqrt{21}}{2} \end{gathered}$$

Evaluate

Find $csct$.

$\begin{array}{rcl}csct& =& \frac{1}{y}\\ csct& =& \frac{1}{{\displaystyle \frac{\sqrt{21}}{5}}}\\ csct& =& \frac{5}{{\displaystyle \sqrt{21}}}\\ csct& =& \frac{5}{{\displaystyle 21}}\sqrt{21}\end{array}$

Find $sect$.

$\begin{array}{rcl}sect& =& \frac{1}{x}\\ sect& =& \frac{1}{-{\displaystyle \frac{2}{5}}}\\ sect& =& -\frac{5}{2}\end{array}$

Find$cott$.

$\begin{array}{rcl}cott& =& \frac{x}{y}\\ cott& =& \frac{-{\displaystyle \frac{2}{5}}}{{\displaystyle \frac{\sqrt{21}}{5}}}\\ cott& =& \frac{-{\displaystyle 2}}{{\displaystyle \sqrt{21}}}\\ cott& =& \frac{-{\displaystyle 2}}{{\displaystyle 21}}\sqrt{21}\end{array}$