Objectives
- Determine the inverse sine, cosine, and tangent values; graph inverse trig functions and determine the limitations on the domain and range.
Section 7.3 Inverse Trig Functions (PF3)
Subsection 7.3.1 Activities
Activity 7.3.1.
Which of the following angles satisfy \(\cos(\theta)=\frac{1}{2}\text{?}\)
- \(\displaystyle \dfrac{\pi}{6}\)
- \(\displaystyle \dfrac{\pi}{3}\)
- \(\displaystyle \dfrac{2\pi}{3}\)
- \(\displaystyle \dfrac{5\pi}{6}\)
- \(\displaystyle \dfrac{7\pi}{6}\)
- \(\displaystyle \dfrac{4\pi}{3}\)
- \(\displaystyle \dfrac{5\pi}{3}\)
- \(\displaystyle \dfrac{11\pi}{6}\)
Answer.
B,G
Activity 7.3.2.
A carpenter is cutting a hand rail for a ramp on his mitre saw. The ramp goes up 4 feet, and the length of the hand rail is 48 feet long. Which of the following equations determines the angle of the ramp, which the carpenter will use to set his saw?
- \(\displaystyle \sin(\theta)=\frac{1}{12}\)
- \(\displaystyle \cos(\theta)=\frac{1}{12}\)
- \(\displaystyle \tan(\theta)=\frac{1}{12}\)
- \(\displaystyle \cot(\theta)=\frac{1}{12}\)
Answer.
A.
Observation 7.3.3.
Often in applications, in addition to finding the sine or cosine of an angle, we need to find an angle with a given sine or cosine or tangent. In other words, we need to have inverse functions of our six trigonometric functions.
Remark 7.3.4.
In Activity 7.3.1, we saw that there are many angles with a given sine or cosine. We must systematically choose one of these to define an inverse function.
Activity 7.3.5.
By restricting the domain, we can find a part of the sine function which is one-to-one, and thus allows us to define an inverse function.
Which of the following domain restrictions is one-to-one?
-
\(0 \leq x \leq \pi\)
-
\(-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}\)
-
\(0 \leq x \leq 2\pi\)
-
\(-\pi \leq x \leq \pi\)
Hint.
Use the horizontal line test.
Answer.
B.
Definition 7.3.6.
The arcsine function, denoted \(\arcsin(x)\text{,}\) is the inverse of the restriction of \(\sin(x)\) to the domain \([-\dfrac{\pi}{2},\dfrac{\pi}{2}]\text{.}\)
In other words, \(\arcsin(x)\) is the unique angle \(\theta\) with \(-\dfrac{\pi}{2} \leq \theta \leq \dfrac{\pi}{2}\) such that \(\sin(\theta)=x\text{.}\)
Activity 7.3.7.
Compute each of the following, without the use of technology.
(a)
\(\arcsin\left(\dfrac{1}{2}\right)\)
Answer.
\(\dfrac{\pi}{6}\)
(b)
\(\arcsin\left(-1\right)\)
Answer.
\(-\dfrac{\pi}{2}\)
(c)
\(\arcsin\left(\dfrac{\sqrt{2}}{2}\right)\)
Answer.
\(\dfrac{\pi}{4}\)
(d)
\(\arcsin\left(-\dfrac{\sqrt{3}}{2}\right)\)
Answer.
\(-\dfrac{\pi}{3}\)
Activity 7.3.8.
Which of the following domain restrictions of \(\cos(x)\) is one-to-one?
-
\(0 \leq x \leq \pi\)
-
\(-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}\)
-
\(0 \leq x \leq 2\pi\)
-
\(-\pi \leq x \leq \pi\)
Hint.
Use the horizontal line test.
Answer.
A.
Definition 7.3.9.
The arccosine function, denoted \(\arccos(x)\text{,}\) is the inverse of the restriction of \(\cos(x)\) to the domain \([0,\pi]\text{.}\)
In other words, \(\arccos(x)\) is the unique angle \(\theta\) with \(0 \leq \theta \leq \pi\) such that \(\cos(\theta)=x\text{.}\)
Activity 7.3.10.
Compute each of the following, without the use of technology.
(a)
\(\arccos\left(\dfrac{1}{2}\right)\)
Answer.
\(\dfrac{\pi}{3}\)
(b)
\(\arccos\left(-1\right)\)
Answer.
\(\pi\)
(c)
\(\arccos\left(\dfrac{\sqrt{2}}{2}\right)\)
Answer.
\(\dfrac{\pi}{4}\)
(d)
\(\arccos\left(-\dfrac{\sqrt{3}}{2}\right)\)
Answer.
\(\frac{5\pi}{6}\)
Activity 7.3.11.
Which of the following domain restrictions of \(\tan(x)\) is one-to-one?
-
\(0 \leq x \leq \pi\)
-
\(-\dfrac{\pi}{2} \leq x \leq \dfrac{\pi}{2}\)
-
\(0 \leq x \leq 2\pi\)
-
\(-\pi \leq x \leq \pi\)
Hint.
Use the horizontal line test.
Answer.
A. and B.
Definition 7.3.12.
The arctangent function, denoted \(\arctan(x)\text{,}\) is the inverse of the restriction of \(\tan(x)\) to the domain \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\text{.}\)
In other words, \(\arctan(x)\) is the unique angle \(\theta\) with \(-\dfrac{\pi}{2} \lt \theta \lt \dfrac{\pi}{2}\) such that \(\tan(\theta)=x\text{.}\)
Observation 7.3.13.
Note that while \(\arcsin(x)\) and \(\arccos(x)\) were defined by restricting the domain to a closed interval, since \(\tan(x)\) is not defined at \(-\dfrac{\pi}{2}\) or \(\dfrac{\pi}{2}\text{,}\) we define \(\arctan(x)\) by restricting the domain to the open interval \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\text{.}\)
Activity 7.3.14.
Compute each of the following, without the use of technology.
(a)
\(\arctan\left(1\right)\)
Answer.
\(\dfrac{\pi}{4}\)
(b)
\(\arctan\left(-\sqrt{3}\right)\)
Answer.
\(-\dfrac{\pi}{3}\)
(c)
\(\arctan\left(0\right)\)
Answer.
\(0\)
(d)
\(\arctan\left(\dfrac{\sqrt{3}}{3}\right)\)
Answer.
\(\dfrac{\pi}{6}\)
Activity 7.3.15.
Sometimes, as in Activity 7.3.2, we need to find an inverse trigonometric function that does not produce one of our special angles.
Compute each of the following using technology (e.g. a calculator).
(a)
\(\arcsin\left(\frac{1}{12}\right)\)
Answer.
\(4.78^\circ\) or \(0.083\) radians
(b)
\(\arccos\left(-\dfrac{3}{5}\right)\)
Answer.
\(126.9^\circ\) or \(2.21\) radians
(c)
\(\arctan\left(2\right)\)
Answer.
\(63.43^\circ\) or \(1.11\) radians
Observation 7.3.16.
Next, we look at the graphs of \(\arcsin(x)\text{,}\) \(\arccos(x)\text{,}\) and \(\arctan(x)\text{.}\)
Activity 7.3.17.
Consider the function \(f(x)=\arcsin(x)\text{.}\)
(a)
Complete the table of values.
| \(x\) | \(\arcsin(x)\) |
|---|---|
| \(-1\) | |
| \(-\dfrac{\sqrt{3}}{2}\) | |
| \(-\dfrac{\sqrt{2}}{2}\) | |
| \(-\dfrac{1}{2}\) | |
| \(0\) | |
| \(\dfrac{1}{2}\) | |
| \(\dfrac{\sqrt{2}}{2}\) | |
| \(\dfrac{\sqrt{3}}{2}\) | |
| \(1\) |
Hint.
Recall that \(\theta=\arcsin(x)\) means \(\sin(\theta)=x\text{.}\)
Answer.
| \(x\) | \(\arcsin(x)\) |
|---|---|
| \(-1\) | \(-\dfrac{\pi}{2}\) |
| \(-\dfrac{\sqrt{3}}{2}\) | \(-\dfrac{\pi}{3}\) |
| \(-\dfrac{\sqrt{2}}{2}\) | \(-\dfrac{\pi}{4}\) |
| \(-\dfrac{1}{2}\) | \(-\dfrac{\pi}{6}\) |
| \(0\) | \(0\) |
| \(\dfrac{1}{2}\) | \(\dfrac{\pi}{6}\) |
| \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\pi}{4}\) |
| \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\pi}{3}\) |
| \(1\) | \(\dfrac{\pi}{2}\) |
(b)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\arcsin(x)\text{.}\) Then sketch the graph of the arcsine curve using the points as a guide.
Answer.
(c)
What is the domain of \(f(x)=\arcsin(x)\text{?}\)
Answer.
\([-1,1]\)
(d)
What is the range of \(f(x)=\arcsin(x)\text{?}\)
Answer.
\([-\dfrac{\pi}{2},\dfrac{\pi}{2}]\)
Activity 7.3.18.
Consider the function \(f(x)=\arccos(x)\text{.}\)
(a)
Complete the table of values.
| \(x\) | \(\arccos(x)\) |
|---|---|
| \(-1\) | |
| \(-\dfrac{\sqrt{3}}{2}\) | |
| \(-\dfrac{\sqrt{2}}{2}\) | |
| \(-\dfrac{1}{2}\) | |
| \(0\) | |
| \(\dfrac{1}{2}\) | |
| \(\dfrac{\sqrt{2}}{2}\) | |
| \(\dfrac{\sqrt{3}}{2}\) | |
| \(1\) |
Hint.
Recall that \(\theta=\arccos(x)\) means \(\cos(\theta)=x\text{.}\)
Answer.
| \(x\) | \(\arccos(x)\) |
|---|---|
| \(-1\) | \(\pi\) |
| \(-\dfrac{\sqrt{3}}{2}\) | \(\dfrac{5\pi}{6}\) |
| \(-\dfrac{\sqrt{2}}{2}\) | \(\dfrac{3\pi}{4}\) |
| \(-\dfrac{1}{2}\) | \(\dfrac{2\pi}{3}\) |
| \(0\) | \(\dfrac{\pi}{2}\) |
| \(\dfrac{1}{2}\) | \(\dfrac{\pi}{3}\) |
| \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\pi}{4}\) |
| \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\pi}{6}\) |
| \(1\) | \(0\) |
(b)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\arccos(x)\text{.}\) Then sketch the graph of the arccosine curve using the points as a guide.
Answer.
(c)
What is the domain of \(f(x)=\arccos(x)\text{?}\)
Answer.
\([-1,1]\)
(d)
What is the range of \(f(x)=\arccos(x)\text{?}\)
Answer.
\([0,\pi]\)
Activity 7.3.19.
Consider the function \(f(x)=\arctan(x)\text{.}\)
(a)
Complete the table of values.
| \(x\) | \(\arctan(x)\) |
|---|---|
| \(-\sqrt{3}\) | |
| \(-1\) | |
| \(-\dfrac{\sqrt{3}}{3}\) | |
| \(0\) | |
| \(\dfrac{\sqrt{3}}{3}\) | |
| \(1\) | |
| \(\sqrt{3}\) |
Hint.
Recall that \(\theta=\arctan(x)\) means \(\tan(\theta)=x\text{.}\)
Answer.
| \(x\) | \(\arctan(x)\) |
|---|---|
| \(-\sqrt{3}\) | \(-\dfrac{\pi}{3}\) |
| \(-1\) | \(-\dfrac{\pi}{4}\) |
| \(-\dfrac{\sqrt{3}}{3}\) | \(-\dfrac{\pi}{6}\) |
| \(0\) | \(0\) |
| \(\dfrac{\sqrt{3}}{3}\) | \(\dfrac{\pi}{6}\) |
| \(1\) | \(\dfrac{\pi}{4}\) |
| \(\sqrt{3}\) | \(\dfrac{\pi}{3}\) |
(b)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\arctan(x)\text{.}\) Then sketch the graph of the arctangent curve using the points as a guide.
Answer.
(c)
What is the domain of \(f(x)=\arctan(x)\text{?}\)
Answer.
\((-\infty,\infty)\)
(d)
What is the range of \(f(x)=\arctan(x)\text{?}\)
Answer.
\(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\)
Activity 7.3.20.
Sometimes when solving applied problems, we need to exactly (not approximately) evaluate expressions like \(\sin\left(\arccos\left(\frac{5}{13}\right)\right)\text{.}\)
(a)
Which of the following sentences describe the expression \(\sin\left(\arccos\left(\frac{5}{13}\right)\right)\text{?}\)
- The angle whose cosine is the same as the sine of \(\frac{5}{13}\text{.}\)
- The angle whose sine is the same as the cosine of \(\frac{5}{13}\text{.}\)
- The cosine of the angle whose sine is \(\frac{5}{13}\text{.}\)
- The sine of the angle whose cosine is \(\frac{5}{13}\text{.}\)
Answer.
D.
(b)
Let \(\theta = \arccos(\frac{5}{13})\text{.}\) Draw a right triangle with an angle of \(\theta\text{,}\) and find the lengths of its three sides.
Answer.
(c)
Find \(\sin(\theta)\text{.}\) Since we defined \(\theta = \arccos(\frac{5}{13})\text{,}\) this gives us \(\sin(\arccos(\frac{5}{13}))\text{.}\)
- \(\displaystyle \dfrac{5}{13}\)
- \(\displaystyle \dfrac{12}{13}\)
- \(\displaystyle \dfrac{5}{12}\)
- \(\displaystyle \dfrac{13}{12}\)
Answer.
B.
Activity 7.3.21.
Compute each of the following.
(a)
\(\tan(\arcsin(\frac{8}{17}))\)
Answer.
\(\dfrac{8}{15}\)
(b)
\(\sec(\arctan(\frac{24}{7}))\)
Answer.
\(\dfrac{25}{7}\)
(c)
\(\tan(\arcsin(\frac{3}{4}))\)
Answer.
\(\dfrac{3}{\sqrt{7}}\)
(d)
\(\cos(\arcsin(x))\)
Hint.
Draw an appropriate right triangle with two sides as \(x\) and \(1\text{.}\)
Answer.
\(\sqrt{1-x^2}\)
Exercises 7.3.2 Exercises
Exercises available at
https://tbil.org/preview/precalculus/exercises/#/bank/PF3/.
