Q. 13 - 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 $t$.

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

Step-by-Step Solution

Answer

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

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

Given information.

The point on the unit circle that corresponds to $t$ is $P\left(x,y\right)=\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$.

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:

Substitute $x=\frac{\sqrt{3}}{2},y=\frac{1}{2}$in the six trigonometric functions and simplify.

Find $\sin t,\cos t,\tan t$.

$\begin{array}{rcl}\sin t& =& y\\ \sin t& =& \frac{1}{2}\end{array}$

$\begin{array}{rcl}\cos t& =& x\\ \cos t& =& \frac{\sqrt{3}}{2}\end{array}$

$\begin{array}{rcl}\tan t& =& \frac{y}{x}\\ \tan t& =& \frac{{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{\sqrt{3}}{2}}}\\ \tan t& =& \frac{1}{\sqrt{3}}\\ \tan t& =& \frac{1}{3}\sqrt{3}\end{array}$

Evaluate

Find $\csc t,\sec t,\cot t$.

$$\begin{gathered} csct=\frac{1}{y} \\ csct=\frac{1}{{\displaystyle \frac{1}{2}}} \\ csct=2 \end{gathered}$$

$\begin{array}{rcl}sect& =& \frac{1}{x}\\ sect& =& \frac{1}{{\displaystyle \frac{\sqrt{3}}{2}}}\\ sect& =& \frac{2}{\sqrt{3}}\\ sect& =& \frac{2}{3}\sqrt{3}\\ \mathrm{co}t t& =& \frac{x}{y}\\ \mathrm{co}t t& =& \frac{{\displaystyle \frac{\sqrt{3}}{2}}}{{\displaystyle \frac{1}{2}}}\\ \mathrm{co}t t& =& \sqrt{3}\end{array}$