1.
Fill in the blanks referring to the angles in Figure 6.3.1.
Section 6.3 Inverse Trigonometry
In this section, we’ll use the inverse sine, cosine, and tangent functions, find the exact value of expressions involving the inverse-trigonometric functions, use a calculator to evaluate inverse-trigonometric functions, and find the exact values of composite functions with inverse-trigonometric functions.
Subsection Textbook Reference
This relates to content in §8.3 of Algebra and Trigonometry 2e.
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.
2.
The invertible function \(g\) is defined by Table 6.3.2. Fill in the \(g^{-1}(x)\) row. Some entries may be "undefined".
| \(x\) | \(-4\) | \(-1\) | \(1\) | \(3\) | \(6\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|
| \(g(x)\) | \(10\) | \(3\) | \(6\) | \(-4\) | \(9\) | \(-1\) | \(1\) |
| \(g^{-1}(x)\) | \(\quad\) | \(\quad\) | \(\quad\) | \(\quad\) | \(\quad\) | \(\quad\) | \(\quad\) |
3.
Show that the graph in Figure 6.3.3 represents an invertible function \(y=R(x)\text{.}\) Graph the inverse \(y = R^{-1}(x)\) alongside \(y=R(x)\text{.}\)
Exercises Practice Exercises
1.
Evaluate the expressions using your memorized values on the unit circle. Some might be undefined.
(a)
\(\sin^{-1}\left(\frac{1}{2}\right)\)
(b)
\(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)
(c)
\(\cos^{-1}\left(1\right)\)
(d)
\(\cos^{-1}\left(\pi\right)\)
(e)
\(\cos^{-1}\left(\frac{1}{2}\right)\)
2.
Fill in Table 6.3.4 using your memorized values on the unit circle.
| \(t\) | \(\quad 1 \quad\) | \(\,\,\frac{\sqrt{3}}{2}\,\,\) | \(\,\,\frac{\sqrt{2}}{2}\,\,\) | \(\quad \frac{1}{2} \quad\) | \(\quad 0 \quad\) | \(\,-\frac{1}{2}\,\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(\,\,-1\,\,\) |
|---|---|---|---|---|---|---|---|---|---|
| \(\cos^{-1}(t)\) | |||||||||
| \(\sin^{-1}(t)\) |
3.
Fill in Table 6.3.5 with standard values of inverse trigonmetric functions without a calculator. If the value of \(z\) is blank, start by filling that in first.
| \(\, z \,\) | \(\sin^{-1}(z)\) | \(\cos^{-1}(z)\) | \(\tan^{-1}(z)\) |
|---|---|---|---|
| \(0\) | |||
| \(-\frac{\pi}{2}\) | |||
| \(\frac{\pi}{6}\) | N/A | ||
| \(\frac{\pi}{4}\) | |||
| \(\frac{\sqrt{2}}{2}\) | N/A | ||
| \(\frac{2\pi}{3}\) | N/A |
4.
Use a calculator to approximate the values below.
(a)
\(\sin^{-1}(0.1)\)
(b)
\(\cos^{-1}(0.6)\)
(c)
\(\tan^{-1}(2)\)
(d)
\(\sec^{-1}(-2)\)
(e)
\(\csc^{-1}(3)\)
5.
Use inverse trigonometry to find the missing angles.
(a)
Use inverse trigonometry and a calculator to find the missing angles \(\alpha\) and \(\beta\) in the triangle shown in Figure 6.3.6. Drawing not to scale.
(b)
Imagine designing a birdhouse to make out of wood. The sides of this birdhouse are shaped like trapezoids, with dimensions shown in Figure 6.3.7. Find the angle \(\phi\text{,}\) in degrees, using inverse trigonometry and your calculator. Note: this angle will be related to how you set up your saw to cut the wood correctly.
A trapezoid showing the side profile of a birdhouse. The base is of length 15cm, the sides are at right angles to the base, the short side is length 20cm, and the long side is length 28cm. The top is of unknown length, but meets the long side at an angle labeled \(\phi\text{.}\)
6.
Evaluate the following expressions without a calculator.
(a)
\(\cos^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right)\)
(b)
\(\cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right)\)
(c)
\(\cos^{-1}\left(\cos\left(-\frac{\pi}{3}\right)\right)\)
(d)
\(\cos^{-1}\left(\cos\left(\frac{6\pi}{7}\right)\right)\)
7.
Evaluate the following expressions without a calculator.
(a)
\(\sin\left(\cos^{-1}\left(\frac{2}{3}\right)\right)\)
(b)
\(\cos\left(\sin^{-1}\left(-\frac{1}{5}\right)\right)\)
(c)
\(\sin\left(\tan^{-1}\left(\frac{4}{7}\right)\right)\)
(d)
\(\cos\left(\tan^{-1}\left(-\frac{7}{2}\right)\right)\)
8.
In this exercise, discuss some of the details about inverse trigonometry with your group members.
(a)
What are the similarities and differences between \(\sin^{-1}(x)\) and \(\csc(x)\text{?}\) Start by discussing the meaning of the inputs and outputs of the functions.
(b)
What are the similarities and differences between \(\sin^{-1}(x)\) and \(\sin^2(x)\text{?}\) Start by discussing the meaning of the inputs and outputs of the functions.
(c)
The range of both \(\sin^{-1}(x)\) and \(\tan^{-1}(x)\) is in quadrants I and IV. The range of \(\cos^{-1}(x)\) is in quadrants I and II. Why do you think that none of the inverse trigonometric functions using quadrant III?
(d)
If Georgiana said that, "I think that the value of \(\sin^{-1}(-1)\) is \(\frac{3\pi}{2}\text{,}\)" how would you help her understand her mistake?
Subsection Definitions
Definition 6.3.8. Inverse Cosine.
The inverse cosine function inputs an \(x\)-value on the unit circle and outputs the angle from \(0\) to \(\pi\) that matches that \(x\)-value, as shown in Figure 6.3.10. Another way to say that is that for angles, \(\theta\text{,}\) in the interval \([0,\pi]\text{,}\) if \(\cos(\theta) = x\) then \(\cos^{-1}(x) = \theta\text{.}\)
| Function | \(\sin\left(z\right)\) | \(\sin^{-1}\left(z\right)\) |
|---|---|---|
| Domain and meaning |
\(\mathbb{R}\) except odd multiples of \(\frac{\pi}{2}\) Angles on unit circle |
\([-1,1]\) \(x\)-values on unit circle |
| Range and meaning |
\([-1,1]\) \(x\)-values on unit circle |
\(\left[0,\pi\right]\) Angles on unit circle |
Definition 6.3.11. Inverse Sine.
The inverse sine function inputs a \(y\)-value on the unit circle and outputs the angle from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) that matches that \(x\)-value, as shown in Figure 6.3.13. Another way to say that is that for angles, \(\theta\text{,}\) in the interval \([-\frac{\pi}{2},\frac{\pi}{2}]\text{,}\) if \(\sin(\theta) = y\) then \(\sin^{-1}(y) = \theta\text{.}\)
| Function | \(\sin\left(z\right)\) | \(\sin^{-1}\left(z\right)\) |
|---|---|---|
| Domain and meaning |
\(\mathbb{R}\) except odd multiples of \(\frac{\pi}{2}\) Angles on unit circle |
\([-1,1]\) \(y\)-values on unit circle |
| Range and meaning |
\([-1,1]\) \(y\)-values on unit circle |
\(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) Angles on unit circle |
Definition 6.3.14. Inverse Tangent.
The inverse tangent function inputs a slope, \(m\text{,}\) and ouptuts the angle from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (non-inclusive) that matches that slope, as shown in Figure 6.3.16. Another way to say that is that for angles, \(\theta\text{,}\) in the interval \([-\frac{\pi}{2},\frac{\pi}{2}]\text{,}\) if \(\tan(\theta) = m\) then \(\tan^{-1}(m) = \theta\text{.}\)
| Function | \(\tan\left(z\right)\) | \(\tan^{-1}\left(z\right)\) |
|---|---|---|
| Domain and meaning |
\(\mathbb{R}\) except odd multiples of \(\frac{\pi}{2}\) Angles on unit circle |
\(\mathbb{R}\) Slopes |
| Range and meaning |
\(\mathbb{R}\) Slopes |
\(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) Angles on unit circle |
Exercises Exit Exercises
1.
Fill in the blanks to show how you should think about the inputs and outputs of each function? The answer to each is either "an angle", "an \(x\)-value on the unit circle", or "\(y\)-value on the unit circle".
(a)
\(\rule{3cm}{0.15mm} = \sin\left(\rule{3cm}{0.15mm}\right)\)
(b)
\(\rule{3cm}{0.15mm} = \cos\left(\rule{3cm}{0.15mm}\right)\)
(c)
\(\rule{3cm}{0.15mm} = \sin^{-1}\left(\rule{3cm}{0.15mm}\right)\)
(d)
\(\rule{3cm}{0.15mm} = \cos^{-1}\left(\rule{3cm}{0.15mm}\right)\)
2.
Evaluate \(\cos^{-1}\left(\cos\left(\frac{5\pi}{4}\right)\right)\text{.}\)
3.
Fill in the missing values in Table 6.3.17.
| \(\quad z \quad\) | \(\sin^{-1}\left(z\right)\) | \(\cos^{-1}\left(z\right)\) |
|---|---|---|
| \(\qquad\) | \(\qquad\) | \(\frac{\pi}{3}\) |
| \(\qquad\) | \(-\frac{\pi}{4}\) | \(\qquad\) |
| \(\frac{-\sqrt{3}}{2}\) | \(\qquad\) | \(\qquad\) |
4.
Use inverse trigonometry and a calculator to find the missing angle \(\zeta\) in the triangle shown in Figure 6.3.18. Drawing is not to scale.
Reflection Reflection
1.
On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
