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Secant, Cosecant, and Cotangent Functions

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Reciprocal Trigonometric Functions

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

Guidance

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

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

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

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

Example A

Use your calculator to evaluate \sec \frac{2 \pi}{5} .

Solution: 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:

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

Example B

Use you calculator to evaluate \cot 100^\circ .

Solution: 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:

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

Example C

Find the exact value of \csc \frac{5 \pi}{3} without using a calculator. Give your answer in exact form.

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

Concept Problem Revisit The secant is the reciprocal of the cosine. So to find \sec 30^\circ , use the cosine.

\cos 30^\circ = \frac{\sqrt{3}}{2} .

Therefore, \sec 30^\circ = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} .

Guided Practice

Use your calculator to evaluate the following reciprocal trigonometric functions.

1. \csc \frac{7 \pi}{8}

2. \cot 85^\circ

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

3. \sec 225^\circ

4. \csc \frac{5 \pi}{6}

Answers

1. \csc \frac{7 \pi}{8}=\frac{1}{\sin \frac{7 \pi}{8}}=2.6131

2. \cot 85^\circ=\frac{1}{\tan 85^\circ}=0.0875

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

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

Vocabulary

Secant
The reciprocal of cosine.
Cosecant
The reciprocal of sine.
Cotangent
The reciprocal of tangent.

Practice

Use your calculator to evaluate the reciprocal trigonometric functions. Round your answers to four decimal places.

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

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

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

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