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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 7.2 Additional Trigonometric Functions (PF2)

Subsection 7.2.1 Activities

Remark 7.2.1.

In the previous sections, we looked at graphs of the sine and cosine functions. We will now look at graphs of the other four trigonometric functions.
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Activity 7.2.2.

Consider the tangent function, \(f(x)=\tan(x)\text{.}\)
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(a)
Fill in the missing values in the table below for \(f(x)=\tan(x)\text{.}\) Find the exact values, then express as a decimal, approximated to two decimal places if needed. (Notice that the values in the table are all the standard angles found on the unit circle!)
\(x\) \(\tan(x)\) (exact) \(\tan(x)\) (as a decimal)
\(0\)
\(\dfrac{\pi}{6}\) \(\dfrac{1}{\sqrt{3}}\)
\(\dfrac{\pi}{4}\)
\(\dfrac{\pi}{3}\) \(\approx 1.73\)
\(\dfrac{\pi}{2}\)
\(\dfrac{2\pi}{3}\)
\(\dfrac{3\pi}{4}\)
\(\dfrac{5\pi}{6}\) \(\approx -0.58\)
\(\pi\)
\(\dfrac{7\pi}{6}\)
\(\dfrac{5\pi}{4}\)
\(\dfrac{4\pi}{3}\) \(\sqrt{3}\)
\(\dfrac{3\pi}{2}\)
\(\dfrac{5\pi}{3}\)
\(\dfrac{7\pi}{4}\)
\(\dfrac{11\pi}{6}\)
\(2\pi\)
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Answer.
\(x\) \(\tan(x)\) (exact) \(\tan(x)\) (as a decimal)
\(0\) \(0\) \(0\)
\(\dfrac{\pi}{6}\) \(\dfrac{1}{\sqrt{3}}\) \(\approx 0.58\)
\(\dfrac{\pi}{4}\) \(1\) \(1\)
\(\dfrac{\pi}{3}\) \(\sqrt{3}\) \(\approx 1.73\)
\(\dfrac{\pi}{2}\) Undefined Undefined
\(\dfrac{2\pi}{3}\) \(-\sqrt{3}\) \(\approx -1.73\)
\(\dfrac{3\pi}{4}\) \(-1\) \(-1\)
\(\dfrac{5\pi}{6}\) \(-\dfrac{1}{\sqrt{3}}\) \(\approx -0.58 \)
\(\pi\) \(0\) \(0\)
\(\dfrac{7\pi}{6}\) \(\dfrac{1}{\sqrt{3}}\) \(\approx 0.58\)
\(\dfrac{5\pi}{4}\) \(1\) \(1\)
\(\dfrac{4\pi}{3}\) \(\sqrt{3}\) \(\approx 1.73\)
\(\dfrac{3\pi}{2}\) Undefined Undefined
\(\dfrac{5\pi}{3}\) \(-\sqrt{3}\) \(\approx -1.73\)
\(\dfrac{7\pi}{4}\) \(-1\) \(-1\)
\(\dfrac{11\pi}{6}\) \(-\dfrac{\sqrt{1}}{3}\) \(\approx-0.58\)
\(2\pi\) \(0\) \(0\)
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(b)
What do you think is happening to the graph at \(\dfrac{\pi}{2}\) and \(\dfrac{3\pi}{2}\text{?}\)
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  1. The graph has a hole.
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  2. The graph has a horizontal asymptote.
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  3. The graph has a vertical asymptote.
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Hint.
Recall that \(\tan(x)=\dfrac{\sin(x)}{\cos(x)}\text{.}\)
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Answer.
C.
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(c)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\tan (x)\text{.}\) Then sketch in the graph of the tangent curve using the points as a guide.
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Answer.
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(d)
What is the domain of \(\tan(x)\text{?}\)
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  1. \(\displaystyle \ldots \cup \left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots\)
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  2. \(\displaystyle \ldots \cup (-\pi,0) \cup (0,\pi) \cup (\pi,2\pi) \cup \ldots\)
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  3. \(\displaystyle \left(0,\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
A.
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(e)
What is the range of \(\tan(x)\text{?}\)
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  1. \(\displaystyle [-1,1]\)
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  2. \(\displaystyle (-\infty,-1] \cup [1,\infty)\)
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  3. \(\displaystyle (-\infty,0) \cup (0,\infty)\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
D.
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(f)
What is the period of \(\tan(x)\text{?}\)
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  1. \(\displaystyle \dfrac{\pi}{2}\)
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  2. \(\displaystyle \pi\)
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  3. \(\displaystyle 2\pi\)
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  4. \(\displaystyle 4\pi\)
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Answer.
B.
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Activity 7.2.3.

Consider the secant function, \(f(x)=\sec(x)\text{.}\)
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(a)
Fill in the missing values in the table below for \(f(x)=\sec(x)\text{.}\) Find the exact values, then express as a decimal, approximated to two decimal places if needed. (Notice that the values in the table are all the standard angles found on the unit circle!)
\(x\) \(\sec(x)\) (exact) \(\sec(x)\) (as a decimal)
\(0\)
\(\dfrac{\pi}{6}\)
\(\dfrac{\pi}{4}\)
\(\dfrac{\pi}{3}\)
\(\dfrac{\pi}{2}\)
\(\dfrac{2\pi}{3}\)
\(\dfrac{3\pi}{4}\) \(\approx -1.41\)
\(\dfrac{5\pi}{6}\)
\(\pi\)
\(\dfrac{7\pi}{6}\) \(-\frac{2}{\sqrt{3}}\)
\(\dfrac{5\pi}{4}\)
\(\dfrac{4\pi}{3}\)
\(\dfrac{3\pi}{2}\)
\(\dfrac{5\pi}{3}\) \(2\)
\(\dfrac{7\pi}{4}\)
\(\dfrac{11\pi}{6}\)
\(2\pi\)
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Answer.
\(x\) \(\sec(x)\) (exact) \(\sec(x)\) (as a decimal)
\(0\) \(1\) \(1\)
\(\dfrac{\pi}{6}\) \(\dfrac{2}{\sqrt{3}}\) \(\approx 1.15\)
\(\dfrac{\pi}{4}\) \(\sqrt{2}\) \(\approx1.41\)
\(\dfrac{\pi}{3}\) \(2\) \(2\)
\(\dfrac{\pi}{2}\) Undefined Undefined
\(\dfrac{2\pi}{3}\) \(-2\) \(-2\)
\(\dfrac{3\pi}{4}\) \(-\sqrt{2}\) \(\approx-1.41\)
\(\dfrac{5\pi}{6}\) \(-\dfrac{2}{\sqrt{3}}\) \(\approx -1.15 \)
\(\pi\) \(-1\) \(-1\)
\(\dfrac{7\pi}{6}\) \(-\dfrac{2}{\sqrt{3}}\) \(\approx -1.15\)
\(\dfrac{5\pi}{4}\) \(-\sqrt{2}\) \(\approx-1.41\)
\(\dfrac{4\pi}{3}\) \(-2\) \(-2\)
\(\dfrac{3\pi}{2}\) Undefined Undefined
\(\dfrac{5\pi}{3}\) \(2\) \(2\)
\(\dfrac{7\pi}{4}\) \(\sqrt{2}\) \(\approx1.41\)
\(\dfrac{11\pi}{6}\) \(\dfrac{2}{\sqrt{3}}\) \(\approx 1.15\)
\(2\pi\) \(1\) \(1\)
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(b)
What do you think is happening to the graph at \(\dfrac{\pi}{2}\) and \(\dfrac{3\pi}{2}\text{?}\)
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  1. The graph has a hole.
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  2. The graph has a horizontal asymptote.
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  3. The graph has a vertical asymptote.
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Answer.
C.
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(c)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\sec (x)\text{.}\) Then sketch in the graph of the secant curve using the points as a guide.
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Answer.
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(d)
What is the domain of \(\sec(x)\text{?}\)
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  1. \(\displaystyle \ldots \cup \left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots\)
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  2. \(\displaystyle \ldots \cup (-\pi,0) \cup (0,\pi) \cup (\pi,2\pi) \cup \ldots\)
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  3. \(\displaystyle \left(0,\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
A.
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(e)
What is the range of \(\sec(x)\text{?}\)
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  1. \(\displaystyle [-1,1]\)
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  2. \(\displaystyle (-\infty,-1] \cup [1,\infty)\)
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  3. \(\displaystyle (-\infty,0) \cup (0,\infty)\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
B.
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(f)
What is the period of \(\sec(x)\text{?}\)
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  1. \(\displaystyle \dfrac{\pi}{2}\)
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  2. \(\displaystyle \pi\)
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  3. \(\displaystyle 2\pi\)
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  4. \(\displaystyle 4\pi\)
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Answer.
C.
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Observation 7.2.4.

Since \(\sec(x)=\dfrac{1}{\cos(x)}\text{,}\) we can see that their graphs are related: \(\sec(x)\) (solid blue curve) has a vertical asymptote everywhere \(\cos(x)\) (dotted green curve) has a zero, and for every point \((a,b)\) on the graph of \(\cos(x)\text{,}\) the point \((a,\frac{1}{b})\) is on the graph of \(\sec(x)\text{.}\)
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Figure 7.2.5. \(y=\sec(x)\)
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Activity 7.2.6.

Consider the cosecant function, \(f(x)=\csc(x)\text{.}\) While we could make a table as in Activity 7.2.3, let’s instead take advantage of the fact that the graphs of \(\csc(x)\) and its reciprocal \(\sin(x)\) will be related in the same way as the graphs of \(\sec(x)\) and its reciprocal \(\cos(x)\text{.}\)
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(a)
Where does \(\sin(x)\) have zeros?
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Hint.
Recall the graph of \(\sin(x)\)
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Figure 7.2.7.
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Answer.
\(\ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\)
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(b)
Where does \(\csc(x)\) have vertical asymptotes?
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Answer.
\(\ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\)
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(c)
Where does \(\sin(x)\) have local maximum and minimum values?
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Answer.
Local maxima are at \(\ldots,-\dfrac{3\pi}{2},\dfrac{\pi}{2},\dfrac{5\pi}{2},\ldots\text{.}\)
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Local minima are at \(\ldots,-\dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{7\pi}{2},\ldots\text{.}\)
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(d)
Where does \(\csc(x)\) have local maximum and minimum values?
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Answer.
Local minima are at \(\ldots,-\dfrac{3\pi}{2},\dfrac{\pi}{2},\dfrac{5\pi}{2},\ldots\text{.}\)
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Local maxima are at \(\ldots,-\dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{7\pi}{2},\ldots\text{.}\)
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(e)
On what intervals is \(\sin(x)\) increasing and decreasing?
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Answer.
Increasing on \(\ldots \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \cup \left(\dfrac{3\pi}{2},\dfrac{5\pi}{2}\right) \cup \ldots \text{.}\)
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Decreasing on \(\ldots \cup \left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots \text{.}\)
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(f)
On what intervals is \(\csc(x)\) increasing and decreasing?
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Answer.
Increasing on \(\ldots \cup \left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots \text{.}\)
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Decreasing on \(\ldots \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \cup \left(\dfrac{3\pi}{2},\dfrac{5\pi}{2}\right) \cup \ldots \text{.}\)
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(g)
Use your answers to the previous tasks to plot the graph of \(y=\csc(x)\text{.}\)
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Hint.
It may be helpful to first sketch the graph of \(\sin(x)\text{.}\)
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Answer.
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(h)
What is the domain of \(\csc(x)\text{?}\)
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  1. \(\displaystyle \ldots \cup \left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots\)
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  2. \(\displaystyle \ldots \cup (-\pi,0) \cup (0,\pi) \cup (\pi,2\pi) \cup \ldots\)
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  3. \(\displaystyle \left(0,\pi\right) \cup \left(\pi,2\pi\right) \cup \ldots\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
B.
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(i)
What is the range of \(\csc(x)\text{?}\)
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  1. \(\displaystyle [-1,1]\)
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  2. \(\displaystyle (-\infty,-1] \cup [1,\infty)\)
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  3. \(\displaystyle (-\infty,0) \cup (0,\infty)\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
B.
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(j)
What is the period of \(\csc(x)\text{?}\)
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  1. \(\displaystyle \dfrac{\pi}{2}\)
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  2. \(\displaystyle \pi\)
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  3. \(\displaystyle 2\pi\)
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  4. \(\displaystyle 4\pi\)
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Answer.
C.
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Observation 7.2.8.

Since \(\csc(x)=\dfrac{1}{\sin(x)}\text{,}\) we can see that their graphs are related: \(\csc(x)\) (solid blue) has a vertical asymptote everywhere \(\sin(x)\) (dotted green) has a zero, and for every point \((a,b)\) on the graph of \(\sin(x)\text{,}\) the point \((a,\frac{1}{b})\) is on the graph of \(\csc(x)\text{.}\)
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Figure 7.2.9. \(y=\csc(x)\)
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Activity 7.2.10.

Consider the cotangent function, \(f(x)=\cot(x)\text{.}\)
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(a)
Where does \(\tan(x)\) have zeros?
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Hint.
Recall the graph of \(\tan(x)\)
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Figure 7.2.11.
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Answer.
\(\ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\)
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(b)
Where does \(\cot(x)\) have vertical asymptotes?
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Answer.
\(\ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\)
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(c)
Where does \(\tan(x)=1\) and where does \(\tan(x)=-1\text{?}\)
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Answer.
\(\tan\left(\dfrac{\pi}{4}+\pi k\right)=1\) for each integer \(k\text{.}\)
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\(\tan\left(\dfrac{3\pi}{4}+\pi k\right)=-1\) for each integer \(k\text{.}\)
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(d)
Where does \(\cot(x)=1\) and where does \(\cot(x)=-1\text{?}\)
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Answer.
\(\cot\left(\dfrac{\pi}{4}+\pi k\right)=1\) for each integer \(k\text{.}\)
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\(\cot\left(\dfrac{3\pi}{4}+\pi k\right)=-1\) for each integer \(k\text{.}\)
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(e)
On what intervals is \(\tan(x)\) increasing and decreasing?
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Answer.
\(\tan(x)\) is increasing everywhere it is defined, and decreasing nowhere.
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(f)
On what intervals is \(\cot(x)\) increasing and decreasing?
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Answer.
\(\cot(x)\) is decreasing everywhere it is defined, and increasing nowhere.
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(g)
Use your answers to the previous tasks to plot the graph of \(y=\cot(x)\text{.}\)
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Hint.
It may be helpful to first sketch the graph of \(\tan(x)\text{.}\)
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Answer.
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(h)
What is the domain of \(\cot(x)\text{?}\)
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  1. \(\displaystyle \ldots \cup \left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right) \cup \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right) \cup \ldots\)
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  2. \(\displaystyle \ldots \cup (-\pi,0) \cup (0,\pi) \cup (\pi,2\pi) \cup \ldots\)
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  3. \(\displaystyle \left(0,\pi\right) \cup \left(\pi,2\pi\right) \cup \ldots\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
B.
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(i)
What is the range of \(\cot(x)\text{?}\)
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  1. \(\displaystyle [-1,1]\)
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  2. \(\displaystyle (-\infty,-1] \cup [1,\infty)\)
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  3. \(\displaystyle (-\infty,0) \cup (0,\infty)\)
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  4. \(\displaystyle (-\infty,\infty)\)
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Answer.
D.
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(j)
What is the period of \(\cot(x)\text{?}\)
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  1. \(\displaystyle \dfrac{\pi}{2}\)
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  2. \(\displaystyle \pi\)
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  3. \(\displaystyle 2\pi\)
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  4. \(\displaystyle 4\pi\)
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Answer.
B.
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Observation 7.2.12.

Since \(\cot(x)=\dfrac{1}{\tan(x)}\text{,}\) we can see that their graphs are related: \(\cot(x)\) has a vertical asymptote everywhere \(\tan(x)\) has a zero (and vice versa), and for every point \((a,b)\) on the graph of \(\tan(x)\text{,}\) the point \((a,\frac{1}{b})\) is on the graph of \(\cot(x)\text{.}\)
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Figure 7.2.13. \(y=\tan(x)\)
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Figure 7.2.14. \(y=\cot(x)\)
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Observation 7.2.16.

Everything we learned about transformations of functions in Section 2.4 applies equally well to trigonometric functions.
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Activity 7.2.17.

Consider the function \(g(x)=\sec\left(x+\dfrac{\pi}{2}\right)\text{.}\)
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(a)
How is this graph related to the graph of \(\sec(x)\text{?}\)
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  1. It is shifted left \(\dfrac{\pi}{2}\text{.}\)
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  2. It is shifted right \(\dfrac{\pi}{2}\text{.}\)
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  3. It is shifted up \(\dfrac{\pi}{2}\text{.}\)
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  4. It is shifted down \(\dfrac{\pi}{2}\text{.}\)
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Answer.
A.
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(c)
What is the domain of \(g(x)=\sec\left(x+\dfrac{\pi}{2}\right)\text{?}\)
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Answer.
\(\ldots \left(-2\pi,-\pi\right) \cup \left(-\pi,0\right)\cup \left(0,\pi\right) \cup \left(\pi,2\pi\right) \cup \ldots \)
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(d)
What is the range of \(g(x)=\sec\left(x+\dfrac{\pi}{2}\right)\text{?}\)
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Answer.
\((-\infty,-1] \cup [1,\infty)\)
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(e)
What is the period of \(g(x)=\sec\left(x+\dfrac{\pi}{2}\right)\text{?}\)
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Answer.
\(2\pi\)
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Activity 7.2.18.

Consider the function \(h(x)=\tan(2x)\)
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(a)
What sort of transformation of \(\tan(x)\) is \(\tan(2x)\text{?}\)
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  1. Horizontal stretch
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  2. Horizontal compression
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  3. Vertical stretch
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  4. Vertical compression
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Answer.
B.
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(b)
Which of the following are different for \(\tan(2x)\) than for \(\tan(x)\text{?}\) Select all that apply.
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  1. Location of vertical asymptotes
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  2. Period
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  3. Domain
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  4. Range
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Answer.
The vertical asymptotes, period, and domain all change, while the range is still \((-\infty,\infty)\text{.}\)
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(c)
Where are the vertical asymptotes of \(h(x)=\tan(2x)\text{?}\)
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  1. \(\displaystyle \ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\)
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  2. \(\displaystyle \ldots,-3\pi,-\pi,\pi,3\pi,\ldots\)
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  3. \(\displaystyle \ldots,-\dfrac{3\pi}{2},-\dfrac{\pi}{2},\dfrac{\pi}{2},\dfrac{3\pi}{2},\ldots\)
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  4. \(\displaystyle \ldots,-\dfrac{3\pi}{4},-\dfrac{\pi}{4},\dfrac{\pi}{4},\dfrac{3\pi}{4},\ldots\)
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Hint.
Recall that \(\tan(x)\) has vertical asymptotes at \(\dfrac{\pi}{2}+\pi k\) for each integer \(k\text{.}\)
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Answer.
D.
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(d)
Where are the zeros of \(h(x)=\tan(2x)\text{?}\)
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  1. \(\displaystyle \ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\)
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  2. \(\displaystyle \ldots,-3\pi,-\pi,\pi,3\pi,\ldots\)
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  3. \(\displaystyle \ldots,,-\dfrac{\pi}{2},0,\dfrac{\pi}{2},\ldots\)
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  4. \(\displaystyle \ldots,-\dfrac{\pi}{4},0,\dfrac{\pi}{4},,\ldots\)
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Hint.
Recall that \(\tan(x)\) has zeroes asymptotes at \(\pi k\) for each integer \(k\text{.}\)
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Answer.
C.
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(f)
What is the domain of \(h(x)=\tan(2x)\text{?}\)
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Answer.
\(\ldots \left(-\dfrac{3\pi}{4},-\dfrac{\pi}{4}\right) \cup \left(-\dfrac{\pi}{4},\dfrac{\pi}{4}\right) \cup \left(\dfrac{\pi}{4},\dfrac{3\pi}{4}\right) \cup \ldots \)
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(g)
What is the range of \(h(x)=\tan(2x)\text{?}\)
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Answer.
\((-\infty,\infty)\)
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(h)
What is the period of \(h(x)=\tan(2x)\text{?}\)
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Answer.
\(\dfrac{\pi}{2}\)
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Activity 7.2.19.

Consider the function \(k(x)=3\csc\left(\dfrac{x}{2}\right)\text{.}\)
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(a)
Where are the vertical asymptotes for \(k(x)=3\csc\left(\dfrac{x}{2}\right)\) located?
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Answer.
\(\ldots,-4\pi,-2\pi,0,2\pi,4\pi,\ldots\)
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(b)
Where are the local minima and local maxima for \(k(x)=3\csc\left(\dfrac{x}{2}\right)\) located?
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Answer.
Local minima are at \(\ldots,-3\pi,\pi,5\pi,\ldots\text{.}\)
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Local maxima are at \(\ldots,-5\pi,-\pi,3\pi,\ldots\text{.}\)
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(c)
What are the local minimum and local maximum values for \(k(x)=3\csc\left(\dfrac{x}{2}\right)\text{?}\)
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Answer.
The local minimum values are \(3\) and the local maximum values are \(-3\)
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(d)
Graph \(k(x)=3\csc\left(\dfrac{x}{2}\right)\text{.}\)
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Answer.
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(e)
What is the domain of \(k(x)=3\csc\left(\dfrac{x}{2}\right)\text{?}\)
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Answer.
\(\ldots \cup (-4\pi,-2\pi) \cup (-2\pi,0) \cup (0,2\pi) \cup (2\pi,4\pi) \cup \ldots\)
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(f)
What is the range of \(k(x)=3\csc\left(\dfrac{x}{2}\right)\text{?}\)
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Answer.
\((-\infty,-3] \cup [3,\infty])\)
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(g)
What is the period of \(k(x)=3\csc\left(\dfrac{x}{2}\right)\text{?}\)
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Answer.
\(4\pi\)
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Subsection 7.2.2 Exercises

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