Chapter 2: Trigonometric Ratios
Exercises: 2.3 Solving Right Triangles
Skills
Suggested Homework Problems
Exercises Homework 2.3
Exercise Group
For Problems 1–4, solve the triangle. Round answers to hundredths.
2.
3.
4.
Exercise Group
For Problems 5–10,
- Sketch the right triangle described.
- Solve the triangle.
5.
[latex]A = 42{^o}, c = 26[/latex]
6.
[latex]B = 28{^o}, c = 6.8[/latex]
7.
[latex]B = 33{^o}, a = 300[/latex]
8.
[latex]B = 79{^o}, a = 116[/latex]
9.
[latex]A = 12{^o}, a = 4[/latex]
10.
[latex]A = 50{^o}, a = 10[/latex]
Exercise Group
For Problems 11–16,
- Sketch the right triangle described.
- Without doing the calculations, list the steps you would use to solve the triangle.
11.
[latex]B = 53.7{^o}, b = 8.2[/latex]
12.
[latex]B = 80{^o}, a = 250[/latex]
13.
[latex]A = 25{^o}, b = 40[/latex]
14.
[latex]A = 15{^o}, c = 62[/latex]
15.
[latex]A = 64.5{^o}, c = 24[/latex]
16.
[latex]B = 44{^o}, b = 0.6[/latex]
Exercise Group
For Problems 17–22, find the labeled angle. Round your answer to tenths of a degree.
17.
18.
19.
20.
21.
22.
Exercise Group
For Problems 23–28, evaluate the expression and sketch a right triangle to illustrate.
23.
[latex]\sin^{-1} 0.2[/latex]
24.
[latex]\cos^{-1} 0.8[/latex]
25.
[latex]\tan^{-1} 1.5[/latex]
26.
[latex]\tan^{-1} 2.5[/latex]
27.
[latex]\cos^{-1} 0.2839[/latex]
28.
[latex]\sin^{-1} 0.4127[/latex]
Exercise Group
For Problems 29–32, write two different equations for the statement.
29.
The cosine of [latex]15 {^o}[/latex] is [latex]0.9659\text{.}[/latex]
30.
The sine of [latex]70 {^o}[/latex] is [latex]0.9397\text{.}[/latex]
31.
The angle whose tangent is [latex]3.1445[/latex] is [latex]65 {^o}\text{.}[/latex]
32.
The angle whose cosine is [latex]0.0872[/latex] is [latex]85 {^o}\text{.}[/latex]
33.
Evaluate the expressions and explain what each means.
[latex]\begin{equation*} \sin^{-1} (0.6), (\sin 6{^o})^{-1} \end{equation*}[/latex]
34.
Evaluate the expressions and explain what each means.
[latex]\begin{equation*} \cos^{-1} (0.36), (\cos 36{^o})^{-1} \end{equation*}[/latex]
Exercise Group
For Problems 35–38,
- Sketch a right triangle that illustrates the situation. Label your sketch with the given information.
- Choose the appropriate trig ratio and write an equation, then solve the problem.
35.
The gondola cable for the ski lift at Snowy Peak is [latex]2458[/latex] yards long and climbs [latex]1860[/latex] feet. What angle with the horizontal does the cable make?
36.
The Leaning Tower of Pisa is [latex]55[/latex] meters in length. An object dropped from the top of the tower lands [latex]4.8[/latex] meters from the base of the tower. At what angle from the horizontal does the tower lean?
37.
A mining company locates a vein of minerals at a depth of [latex]32[/latex] meters. However, there is a layer of granite directly above the minerals, so they decide to drill at an angle, starting [latex]10[/latex] meters from their original location. At what angle from the horizontal should they drill?
38.
The birdhouse in Carolyn’s front yard is [latex]12[/latex] feet tall, and its shadow at [latex]4[/latex] pm is [latex]15[/latex] feet [latex]4[/latex] inches long. What is the angle of elevation of the sun at [latex]4[/latex] pm?
Exercise Group
For Problems 39–42,
- Sketch the right triangle described.
- Solve the triangle.
39.
[latex]a = 18, b = 26[/latex]
40.
[latex]a = 35, b = 27[/latex]
41.
[latex]b = 10.6 , c = 19.2[/latex]
42.
[latex]a = 88, c = 132[/latex]
Exercise Group
For Problems 43–48,
- Make a sketch that illustrates the situation. Label your sketch with the given information.
- Write an equation and solve the problem.
43.
The Mayan pyramid of El Castillo at Chichén Itzá in Mexico has [latex]91[/latex] steps. Each step is 26 cm high and 30 cm deep.
- What angle does the side of the pyramid make with the horizontal?
- What is the distance up the face of the pyramid, from base to top platform?
44.
An airplane begins its descent when its altitude is 10 kilometers. The angle of descent should be [latex]3{^o}[/latex] from horizontal.
- How far from the airport (measured along the ground) should the airplane begin its descent?
- How far will the airplane travel on its descent to the airport?
45.
A communications satellite is in a low earth orbit (LOE) at an altitude of 400 km. From the satellite, the angle of depression to earth’s horizon is [latex]19.728{^o}\text{.}[/latex] Use this information to calculate the radius of the earth.
46.
The first Ferris wheel was built for the [latex]1893[/latex] Chicago World’s Fair. It had a diameter of [latex]250[/latex] feet, and the boarding platform, at the base of the wheel, was [latex]14[/latex] feet above the ground. If you boarded the wheel and rotated through an angle of [latex]50{^o}\text{,}[/latex] what would be your height above the ground?
47.
To find the distance across a ravine, Delbert takes some measurements from a small airplane. When he is a short distance from the ravine at an altitude of [latex]500[/latex] feet, he finds that the angle of depression to the near side of the ravine is [latex]56{^o}\text{,}[/latex] and the angle of depression to the far side is [latex]32{^o}\text{.}[/latex] What is the width of the ravine? (Hint: First find the horizontal distance from Delbert to the near side of the ravine.)
48.
The window in Francine’s office is [latex]4[/latex] feet wide and [latex]5[/latex] feet tall. The bottom of the window is 3 feet from the floor. When the sun is at an angle of elevation of [latex]64{^o}\text{,}[/latex] what is the area of the sunny spot on the floor?
49.
Which of the following numbers are equal to [latex]\cos 45{^o}\text{?}[/latex]
- [latex]\displaystyle \dfrac{\sqrt{2}}{2}[/latex]
- [latex]\displaystyle \dfrac{1}{\sqrt{2}}[/latex]
- [latex]\displaystyle \dfrac{2}{\sqrt{2}}[/latex]
- [latex]\displaystyle \sqrt{2}[/latex]
50.
Which of the following numbers are equal to [latex]\tan 30{^o}\text{?}[/latex]
- [latex]\displaystyle \sqrt{3}[/latex]
- [latex]\displaystyle \dfrac{1}{\sqrt{3}}[/latex]
- [latex]\displaystyle \dfrac{\sqrt{3}}{3}[/latex]
- [latex]\displaystyle \dfrac{3}{\sqrt{3}}[/latex]
51.
Which of the following numbers are equal to [latex]\tan 60{^o}\text{?}[/latex]
- [latex]\displaystyle \sqrt{3}[/latex]
- [latex]\displaystyle \dfrac{1}{\sqrt{3}}[/latex]
- [latex]\displaystyle \dfrac{\sqrt{3}}{3}[/latex]
- [latex]\displaystyle \dfrac{3}{\sqrt{3}}[/latex]
52.
Which of the following numbers are equal to [latex]\sin 60{^o}\text{?}[/latex]
- [latex]\displaystyle \dfrac{3}{\sqrt{2}}[/latex]
- [latex]\displaystyle \dfrac{\sqrt{3}}{2}[/latex]
- [latex]\displaystyle \dfrac{\sqrt{2}}{3}[/latex]
- [latex]\displaystyle \dfrac{2}{\sqrt{3}}[/latex]
Exercise Group
For Problems 53–58, choose all values from the list below that are exactly equal to, or decimal approximations for, the given trig ratio. (Try not to use a calculator!)
| [latex]\sin 30{^o}[/latex] | [latex]\cos 45{^o}[/latex] | [latex]\sin 60{^o}[/latex] | [latex]\tan 45{^o}[/latex] | [latex]\tan 60{^o}[/latex] |
| [latex]0.5000[/latex] | [latex]0.5774[/latex] | [latex]0.7071[/latex] | [latex]0.8660[/latex] | [latex]1.0000[/latex] |
| [latex]\dfrac{1}{\sqrt{2}}[/latex] | [latex]\dfrac{2}{\sqrt{2}}[/latex] | [latex]\dfrac{3}{\sqrt{2}}[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]\dfrac{\sqrt{2}}{2}[/latex] |
| [latex]\dfrac{1}{\sqrt{3}}[/latex] | [latex]\dfrac{2}{\sqrt{3}}[/latex] | [latex]\dfrac{\sqrt{3}}{2}[/latex] | [latex]\sqrt{3}[/latex] | [latex]\dfrac{\sqrt{3}}{3}[/latex] |
53.
[latex]\cos 30{^o}[/latex]
54.
[latex]\sin 45{^o}[/latex]
55.
[latex]\tan 30{^o}[/latex]
56.
[latex]\cos 60{^o}[/latex]
57.
[latex]\sin 90{^o}[/latex]
58.
[latex]\cos 0{^o}[/latex]
59.
Fill in the table from memory with exact values. Do you notice any patterns that might help you memorize the values?
| [latex]\theta[/latex] | [latex]0{^o}[/latex] | [latex]30{^o}[/latex] | [latex]45{^o}[/latex] | [latex]60{^o}[/latex] | [latex]90{^o}[/latex] |
| [latex]\sin \theta[/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] |
| [latex]\cos \theta[/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] |
| [latex]\tan \theta[/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] |
60.
Fill in the table from memory with decimal approximations to four places.
| [latex]\theta[/latex] | [latex]0{^o}[/latex] | [latex]30{^o}[/latex] | [latex]45{^o}[/latex] | [latex]60{^o}[/latex] | [latex]90{^o}[/latex] |
| [latex]\sin \theta[/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] |
| [latex]\cos \theta[/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] |
| [latex]\tan \theta[/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] | [latex][/latex] |
Exercise Group
For Problems 61 and 62, compare the given value with the trig ratios of the special angles to answer the questions. Try not to use a calculator.
61.
Is the acute angle larger or smaller than [latex]45{^o}\text{?}[/latex]
- [latex]\displaystyle \sin \alpha = 0.7[/latex]
- [latex]\displaystyle \tan \beta = 1.2[/latex]
- [latex]\displaystyle \cos \gamma = 0.65[/latex]
62.
Is the acute angle larger or smaller than [latex]60{^o}\text{?}[/latex]
- [latex]\displaystyle \cos \theta = 0.75[/latex]
- [latex]\displaystyle \tan \phi = 1.5[/latex]
- [latex]\displaystyle \sin \psi = 0.72[/latex]
Exercise Group
For Problems 63–72, solve the triangle. Give your answers as exact values.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
- Find the perimeter of a regular hexagon if the apothegm is [latex]8[/latex] cm long. (The apothegm is the segment from the center of the hexagon and perpendicular to one of its sides.)
- Find the area of the hexagon.
74.
Triangle [latex]ABC[/latex] is equilateral, and its angle bisectors meet at point [latex]P\text{.}[/latex] The sides of [latex]\triangle ABC[/latex] are 6 inches long. Find the length of [latex]AP\text{.}[/latex]
75.
Find an exact value for the area of each triangle.
76.
Find an exact value for the perimeter of each parallelogram.
77.
- Find the area of the outer square.
- Find the dimensions and the area of the inner square.
- What is the ratio of the area of the outer square to the area of the inner square?
78.
- Find the area of the inner square.
- Find the dimensions and the area of the outer square.
- What is the ratio of the area of the outer square to the area of the inner square?