Other Trigonometric Functions

Section 5.4 Other Trigonometric Functions

In this section, we’ll find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of \(\frac{\pi}{3}\text{,}\) \(\frac{\pi}{4}\text{,}\) and \(\frac{\pi}{3}\text{,}\) use reference angles to evaluate the trigonometric functions secant, tangent, and cotangent, use properties of even and odd trigonometric functions, recognize and use fundamental identities, and evaluate trigonometric functions with a calculator.

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Subsection Textbook Reference

This relates to content in §7.4 of Algebra and Trigonometry 2e.

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Exercises Preparation Exercises

Answer the following without using a calculator. These questions are intended to check the prerequisite skills needed to complete the rest of the lab.

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1.

What does "CHOSHACAO" stand for? See Right Angle Trigonometry Definitions .

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2.

What is the value of \(\sin(\frac{\pi}{3])\text{?}\)

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3.

Find the value of \(\cos(\frac{11\pi}{4])\) using your memorized unit circle values and a reference angle.

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4.

Find the value of \(\sin(-\frac{2\pi}{3])\) using your memorized unit circle values and a reference angle.

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5.

Explain what makes a function an "odd function"?

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Exercises Practice Exercises

1.

Find all six trigonometric function values given the coordinates at an angle \(\gamma\) on a unit circle.

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described in detail following the image — Figure 5.4.1. A point on a unit circle at an angle \(\gamma\)
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(a)

\(\sin(\gamma)\)

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(b)

\(\cos(\gamma)\)

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(c)

\(\csc(\gamma)\)

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(d)

\(\sec(\gamma)\)

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(e)

\(\tan(\gamma)\)

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(f)

\(\cot(\gamma)\)

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2.

Find all six trigonometric function values at the angle \(\frac{\pi}{3}\) using your memorized values for \(\sin\left(\frac{\pi}{3}\right)\) and \(\cos\left(\frac{\pi}{3}\right)\text{.}\)

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(a)

\(\sin(\frac{\pi}{3})\)

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(b)

\(\cos(\frac{\pi}{3})\)

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(c)

\(\csc(\frac{\pi}{3})\)

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(d)

\(\sec(\frac{\pi}{3})\)

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(e)

\(\tan(\frac{\pi}{3})\)

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(f)

\(\cot(\frac{\pi}{3})\)

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3.

Find the trigonometric function values given the right triangle with lengths shown in Figure 5.4.2.

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(a)

\(\sin(\theta)\)

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(b)

\(\csc(\theta)\)

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(c)

\(\sec(\theta)\)

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(d)

\(\cot(\theta)\)

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4.

Fill in a table of all 6 trig values for the standard angles on the unit circle.

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\(\theta\)	\(\quad 0 \quad\)	\(\quad\frac{5\pi}{6}\quad\)	\(\quad\frac{\pi}{4}\quad\)	\(\quad\frac{4\pi}{3}\quad\)	\(\quad\frac{\pi}{2}\quad\)
\(\sin(\theta)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)
\(\cos(\theta)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)
\(\csc(\theta)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)
\(\sec(\theta)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)
\(\tan(\theta)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)
\(\cot(\theta)\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)	\(\qquad\)

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5.

Use the fact that \(\sin(x)\text{,}\) \(\tan(x)\text{,}\) \(\cot(x)\text{,}\) and \(\csc(x)\) are all odd, and \(\cos(x)\) and \(\sec(x)\) are even to find the following values.

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(a)

\(\sin\left(-\frac{2\pi}{3}\right)\)

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(b)

\(\tan\left(-\frac{2\pi}{3}\right)\)

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(c)

\(\cot\left(-\frac{2\pi}{3}\right)\)

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(d)

\(\csc\left(-\frac{2\pi}{3}\right)\)

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(e)

\(\cos\left(-\frac{2\pi}{3}\right)\)

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(f)

\(\sec\left(-\frac{2\pi}{3}\right)\)

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6.

Find the other five trigonometric values for the angle shown in each part.

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(a)

\(\sec(\psi) = -\frac{7}{2}\) and \(\frac{\pi}{2} \leq \psi \leq \pi\text{.}\)

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(b)

\(\cot(\zeta) = -\frac{6}{5}\) and \(\frac{3\pi}{2} \leq \zeta \leq 2\pi\text{.}\)

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Subsection Definitions

Definition 5.4.3. The "Other" Trigonometric Functions.

For a right triangle with angle \(\theta\text{,}\) the "other trig functions" can be found by the following definitions as in Figure 5.4.4. Visit this interactive Desmos graph to see the six trigonometric lengths in action.

\begin{gather*} \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\\ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\\ \sec(\theta) = \frac{1}{\cos(\theta)}\\ \csc(\theta) = \frac{1}{\sin(\theta)} \end{gather*}

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Definition 5.4.5. CHOSHACAO.

For a right triangle with hypotenuse \(h\text{,}\) and sides \(x\) and \(y\text{,}\) as in Figure 5.4.6, tangent is still "opposite over adjacent", as you saw in the section about right-angle trigonometry and the "other trigonmetric function values" can be found with "CHOSHACAO". Visit this interactive Desmos graph to see the six trigonometric lengths in action.

\begin{gather*} \tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}\\ \cot(\theta) = \frac{x}{y} = \frac{\cos(\theta)}{\sin(\theta)}\\ \sec(\theta) = \frac{h}{x} = \frac{1}{\cos(\theta)}\\ \csc(\theta) = \frac{h}{y} = \frac{1}{\sin(\theta)} \end{gather*}

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Definition 5.4.7. Even and Odd Trigonometric Functions.

Recall that even functions are symmetrical about the \(y\)-axis, which makes \(f(x)=f(-x)\text{,}\) and odd functions are symmetrical about the origin, which makes \(-f(x) = f(-x)\text{.}\) All of the trigonometric functions are either even or odd. Here’s a list:

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\(\sin(-x) = -\sin(x)\text{,}\) so \(\sin(x)\) is odd.
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\(\tan(-x) = -\tan(x)\text{,}\) so \(\tan(x)\) is odd.
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\(\cot(-x) = -\cot(x)\text{,}\) so \(\cot(x)\) is odd.
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\(\csc(-x) = -\csc(x)\text{,}\) so \(\csc(x)\) is odd.
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\(\cos(-x) = \cos(x)\text{,}\) so \(\cos(x)\) is even.
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\(\sec(-x) = \sec(x)\text{,}\) so \(\sec(x)\) is even.
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Definition 5.4.8. Other Pythagorean Identities.

You already know that \(\sin^2(\theta) + \cos^2(\theta) = 1\text{,}\) and that this is called the Pythagorean Identity. There are two other Pythagorean relationships which you can see in Figure 5.4.9, which is a trimmed down version of Figure 5.4.4.

\begin{gather*} \tan^2(\theta) + 1 = \sec^2(\theta)\\ \cot^2(\theta) + 1 = \csc^2(\theta) \end{gather*}

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Definition 5.4.10. Periods of Trigonometric Functions.

The period of a function is a length of the shortest \(x\)-interval over which a function completes one full cycle. This can be thought of as the shortest distance that a graph can be shifted left or right before completely aligning with the original graph. Mathematically, this would be that the period, \(P\text{,}\) of a repeating function \(f\) is the smallestvalue such that \(f(x+P) = f(x)\text{,}\) for any value of \(x\text{.}\)

The period of \(\sin(x)\text{,}\) \(\csc(x)\text{,}\) \(\cos(x)\text{,}\) and \(\sec(x)\) is \(2\pi\text{.}\)
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The period of \(\tan(x)\) and \(\cot(x)\) is \(\pi\text{.}\)
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