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Secant, Cosecant, and Cotangent Functions

Secant, cosecant, cotangent values of common angles

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Practice Secant, Cosecant, and Cotangent Functions

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Reciprocal Trigonometric Functions

A ladder propped up against a house forms an angle of \begin{align*}30^\circ\end{align*} with the ground. What is the secant of this angle?

Reciprocal Trigonometric Functions

Each of the trigonometric ratios has a reciprocal function associated with it as shown below.

The reciprocal of sine is cosecant: \begin{align*}\frac{1}{\sin \theta}=\csc \theta\end{align*}, so \begin{align*}\csc \theta=\frac{H}{O}\end{align*} (hypotenuse over opposite)

The reciprocal of cosine is secant: \begin{align*}\frac{1}{\cos \theta}=\sec \theta\end{align*}, so \begin{align*}\sec \theta=\frac{H}{A}\end{align*} (hypotenuse over adjacent)

The reciprocal of tangent is cotangent: \begin{align*}\frac{1}{\tan \theta}=\cot \theta\end{align*}, so \begin{align*}\cot \theta=\frac{A}{O}\end{align*} (adjacent over opposite)

Let's use a calculator to evaluate \begin{align*}\sec \frac{2 \pi}{5}\end{align*}.

First, be sure that your calculator is in radian mode. To check/change the mode, press the MODE button and make sure RADIAN is highlighted. If it is not, use the arrow keys to move the cursor to RADIANS and press enter to select RADIAN as the mode. Now we are ready to use the calculator to evaluate the reciprocal trig function. Since the calculator does not have a button for secant, however, we must utilize the reciprocal relationship between cosine and secant:

\begin{align*}\text{Since}\ \sec \theta=\frac{1}{\cos \theta}, \sec \frac{2 \pi}{5}=\frac{1}{\cos \frac{2 \pi}{5}}=3.2361.\end{align*}

Now, let's use a calculator to evaluate \begin{align*}\cot 100^\circ\end{align*}.

This time we will need to be in degree mode. After the mode has been changed we can use the reciprocal of cotangent, which is tangent, to evaluate as shown:

\begin{align*}\text{Since}\ \cot \theta=\frac{1}{\tan \theta}, \cot 100^\circ=\frac{1}{\tan 100^\circ} \approx -0.1763.\end{align*}

Finally, let's find the exact value of \begin{align*}\csc \frac{5 \pi}{3}\end{align*} without using a calculator and give our answer in exact form.

The reciprocal of cosecant is sine so we will first find \begin{align*}\sin \frac{5 \pi}{3}\end{align*} Using either the unit circle or the alternative method, we can determine that \begin{align*}\sin \frac{5 \pi}{3}\end{align*} is \begin{align*}-\frac{\sqrt{3}}{2}\end{align*} using a \begin{align*}60^\circ\end{align*} reference angle in the fourth quadrant. Now, find its reciprocal: \begin{align*}\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}=-\frac{2 \sqrt{3}}{3}\end{align*}.

Examples

Example 1

Earlier, you were asked to find the secant of the angle of the ladder propped up against the house.

The secant is the reciprocal of the cosine. So to find \begin{align*}\sec 30^\circ\end{align*}, use the cosine.

\begin{align*}\cos 30^\circ = \frac{\sqrt{3}}{2}\end{align*}.

Therefore, \begin{align*}\sec 30^\circ = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\end{align*}.

Use your calculator to evaluate the following reciprocal trigonometric functions.

Example 2

\begin{align*}\csc \frac{7 \pi}{8}\end{align*}

\begin{align*}\csc \frac{7 \pi}{8}=\frac{1}{\sin \frac{7 \pi}{8}}=2.6131\end{align*}

Example 3

\begin{align*}\cot 85^\circ\end{align*}

\begin{align*}\cot 85^\circ=\frac{1}{\tan 85^\circ}=0.0875\end{align*}

Evaluate the following without using a calculator. Give all answers in exact form.

Example 4

\begin{align*}\sec 225^\circ\end{align*}

\begin{align*}\sec 225^\circ\end{align*} is the reciprocal of \begin{align*}\cot 225^\circ\end{align*}, a \begin{align*}45^\circ\end{align*} reference angle in quadrant three where cosine is negative. Because \begin{align*}\cos 45^\circ=\frac{1}{\sqrt{2}}, \cos 225^\circ=-\frac{1}{\sqrt{2}}\end{align*}, and \begin{align*}\sec 225^\circ=-\sqrt{2}\end{align*}.

Example 5

\begin{align*}\csc \frac{5 \pi}{6}\end{align*}

\begin{align*}\csc \frac{5 \pi}{6}\end{align*} is the reciprocal of \begin{align*}\sin \frac{5 \pi}{6}\end{align*}, a \begin{align*}\frac{\pi}{6}\end{align*} or \begin{align*}30^\circ\end{align*} reference angle in the second quadrant where sine is positive. Because \begin{align*}\sin \frac{\pi}{6}=\frac{1}{2}, \sin \frac{5 \pi}{6}=\frac{1}{2},\end{align*} and \begin{align*}\csc \frac{5 \pi}{6}=2\end{align*}.

Review

1. \begin{align*}\csc 95^\circ\end{align*}
2. \begin{align*}\cot 278^\circ\end{align*}
3. \begin{align*}\sec \frac{14 \pi}{5}\end{align*}
4. \begin{align*}\cot(-245^\circ)\end{align*}
5. \begin{align*}\sec \frac{6 \pi}{7}\end{align*}
6. \begin{align*}\csc \frac{23 \pi}{13}\end{align*}
7. \begin{align*}\cot 333^\circ\end{align*}
8. \begin{align*}\csc \frac{9 \pi}{5}\end{align*}

Evaluate the following trigonometric functions without using a calculator. Give your answers exactly.

1. \begin{align*}\sec \frac{5 \pi}{6}\end{align*}
2. \begin{align*}\csc \left(-\frac{3 \pi}{2}\right)\end{align*}
3. \begin{align*}\cot 225^\circ\end{align*}
4. \begin{align*}\sec \frac{11 \pi}{3}\end{align*}
5. \begin{align*}\csc \frac{7 \pi}{6}\end{align*}
6. \begin{align*}\sec 270^\circ\end{align*}
7. \begin{align*}\cot \frac{5 \pi}{3}\end{align*}
8. \begin{align*}\csc 315^\circ\end{align*}

To see the Review answers, open this PDF file and look for section 13.8.

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