Suggested languages for you:

Americas

Europe

Q. 14

Expert-verified
Found in: Page 380

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

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

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

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

See the 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:

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

## Evaluate

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

Substitute these values in the six trigonometric functions and simplify.

Find $\mathrm{sin}t$.

$\begin{array}{rcl}\mathrm{sin}t& =& y\\ \mathrm{sin}t& =& -\frac{\sqrt{3}}{2}\end{array}$

Find role="math" localid="1646311483257" $\mathrm{cos}t$.

$\begin{array}{rcl}\mathrm{cos}t& =& x\\ \mathrm{cos}t& =& \frac{1}{2}\end{array}$

Find $\mathrm{tan}t$.

## Evaluate.

Find $csct$.

Find $sect$.

Find $cott$.