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Q. 49

Expert-verified
Found in: Page 381

### Precalculus Enhanced with Graphing Utilities

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

# Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator.$210°$

The exact values of the six trigonometric functions are $\mathrm{sin}\left(210°\right)=-\frac{1}{2}$, $\mathrm{cos}\left(210°\right)=-\frac{\sqrt{3}}{2}$, $\mathrm{tan}\left(210°\right)=\frac{1}{\sqrt{3}}$, $\mathrm{csc}\left(210°\right)=-2$, $\mathrm{sec}\left(210°\right)=-\frac{2}{\sqrt{3}}$ and $\mathrm{cot}\left(210°\right)=\sqrt{3}$.

See the step by step solution

## Step 1. Determine a point corresponds to the given angle.

The given angle is $210°=\frac{7\pi }{6}$. This angle is multiple of $\frac{\pi }{6}$.

From the below figure, we see the point $\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$ corresponds to $210°$.

## Step 2. Determine the exact value of sine function.

The exact value of the sine function is:

$\mathrm{sin}\left(210°\right)=y\phantom{\rule{0ex}{0ex}}\mathrm{sin}\left(210°\right)=-\frac{1}{2}$

## Step 3. Determine the exact value of cosine function.

The exact value of the cosine function is:

$\mathrm{cos}\left(210°\right)=x\phantom{\rule{0ex}{0ex}}\mathrm{cos}\left(210°\right)=-\frac{\sqrt{3}}{2}$

## Step 4. Determine the exact value of tangent function.

The exact value of the tangent function is:

$\begin{array}{rcl}\mathrm{tan}\theta & =& \frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }\\ \mathrm{tan}\left(210°\right)& =& \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\\ \mathrm{tan}\left(210°\right)& =& \frac{1}{\sqrt{3}}\end{array}$

## Step 5. Determine the exact value of cosecant function.

The exact value of the cosecant function is:

$\begin{array}{rcl}\mathrm{csc}\theta & =& \frac{1}{\mathrm{sin}\theta }\\ \mathrm{csc}\left(210°\right)& =& \frac{1}{-\frac{1}{2}}\\ \mathrm{csc}\left(210°\right)& =& -2\end{array}$

## Step 6. Determine the exact value of secant function.

The exact value of the secant function is:

$\begin{array}{rcl}\mathrm{sec}\theta & =& \frac{1}{\mathrm{cos}\theta }\\ \mathrm{sec}\left(210°\right)& =& \frac{1}{\frac{-\sqrt{3}}{2}}\\ \mathrm{sec}\left(210°\right)& =& -\frac{2}{\sqrt{3}}\end{array}$

## Step 7. Determine the exact value of cotangent function.

The exact value of the cotangent function is:

$\begin{array}{rcl}\mathrm{cot}\theta & =& \frac{1}{\mathrm{tan}\theta }\\ \mathrm{cot}\left(210°\right)& =& \frac{1}{\frac{1}{\sqrt{3}}}\\ \mathrm{cot}\left(210°\right)& =& \sqrt{3}\end{array}$