Exercises: 2.2 Right Triangle Trigonometry
Exercises Homework 2.2
Skills
-
- Use measurements to calculate the trigonometric ratios for acute angles #1-10, 57-60
- Use trigonometric ratios to find unknown sides of right triangles #11-26
- Solve problems using trigonometric ratios #27-34, 41-46
- Use trig ratios to write equations relating the sides of a right triangle #35-40
- Use relationships among the trigonometric ratios #47-56, 61-68
Problems: 6, 10, 58, 12, 22, 30, 42, 46, 38, 40, 50, 62.
1.
Here are two right triangles with a [latex]65 °[/latex] angle.
- Measure the sides [latex]AB[/latex] and [latex]BC[/latex] with a ruler. Use the lengths to estimate [latex]\sin 65°{.}[/latex]
- Measure the sides [latex]AD[/latex] and [latex]DE[/latex] with a ruler. Use the lengths to estimate [latex]\sin 65°{.}[/latex]
- Use your calculator to look up [latex]\sin 65°{.}[/latex] Compare your answers. How close were your estimates?
2.
Use the figure in Problem 1 to calculate two estimates each for the cosine and tangent of [latex]65 °{.}[/latex] Compare your estimates to your calculator’s values for [latex]\cos 65°[/latex] and [latex]\tan 65°{.}[/latex]
3.
Here are two right triangles with a [latex]40 °[/latex] angle.
- Measure the sides [latex]AB[/latex] and [latex]AC[/latex] with a ruler. Use the lengths to estimate [latex]\cos 40°{.}[/latex]
- Measure the sides [latex]AD[/latex] and [latex]AE[/latex] with a ruler. Use the lengths to estimate [latex]\cos 40°{.}[/latex]
- Use your calculator to look up [latex]\cos 40°{.}[/latex] Compare your answers. How close were your estimates?
4.
Use the figure in Problem 2 to calculate two estimates each for the cosine and tangent of [latex]40 °{.}[/latex] Compare your estimates to your calculator’s values for [latex]\sin 40°[/latex] and [latex]\tan 40°{.}[/latex]
Exercise Group
For the right triangles in Problems 5–10,
- Find the length of the unknown side.
- Find the sine, cosine, and tangent of [latex]\theta \text{.}[/latex] Round your answers to four decimal places.
5.
6.
7.
8.
9.
10.
Exercise Group
For Problems 11–16,
- Sketch and label the sides of a right triangle with angle [latex]\theta\text{.}[/latex]
- Sketch and label another right triangle with angle [latex]\theta[/latex] and longer sides.
11.
[latex]\cos \theta = \dfrac{3}{5}[/latex]
12.
[latex]\tan \theta = \dfrac{7}{2}[/latex]
13.
[latex]\tan \theta = \dfrac{11}{4}[/latex]
14.
[latex]\sin \theta = \dfrac{4}{9}[/latex]
15.
[latex]\sin \theta = \dfrac{1}{9}[/latex]
16.
[latex]\cos \theta = \dfrac{7}{8}[/latex]
Exercise Group
For Problems 17–22, use one of the three trigonometric ratios to find the unknown side of the triangle. Round your answer to hundredths.
17.
18.
19.
20.
21.
22.
Exercise Group
For Problems 23–26, sketch and label a right triangle with the given properties.
23.
One angle is [latex]40°{,}[/latex] the side opposite that angle is 8 inches
24.
One angle is [latex]65°{,}[/latex] the side adjacent to that angle is 30 yards
25.
One angle is [latex]28°{,}[/latex] the hypotenuse is 56 feet
26.
One leg is [latex]15[/latex] meters, the hypotenuse is [latex]18[/latex] meters
Exercise Group
For Problems 27–34,
- 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.
27.
To measure the height of cloud cover, airport controllers fix a searchlight to shine a vertical beam on the clouds. The searchlight is [latex]120[/latex] yards from the office. A technician in the office measures the angle of elevation to the light on the cloud cover at [latex]54.8°{.}[/latex] What is the height of the cloud cover?
28.
To measure the distance across a canyon, Evel first sights an interesting rock directly opposite on the other side. He then walks [latex]200[/latex] yards down the rim of the canyon and sights the rock again, this time at an angle of [latex]18.5°[/latex] from the canyon rim. What is the width of the canyon?
29.
A salvage ship is searching for the wreck of a pirate vessel on the ocean floor. Using sonar, they locate the wreck at an angle of depression of [latex]36.2°{.}[/latex] The depth of the ocean at their location is [latex]260[/latex] feet. How far should they move so that they are directly above the wrecked vessel?
30.
Ramps for wheelchairs should be no steeper than an angle of [latex]6°{.}[/latex] How much horizontal distance should be allowed for a ramp that rises [latex]5[/latex] feet in height?
31.
The radio signal from a weather balloon indicates that it is [latex]1500[/latex] meters from a meteorologist on the ground. The angle of elevation to the balloon is [latex]48°{.}[/latex] What is the balloon’s altitude?
32.
According to Chinese legend, around 200 BC, the general Han Xin used a kite to determine the distance from his location to an enemy palace. He then dug a secret tunnel which emerged inside the palace. When the kite was directly above the palace, its angle of elevation was [latex]27°[/latex] and the string to the kite was [latex]1850[/latex] feet long. How far did Han Xin’s troops have to dig?
33.
A cable car on a ski lift traverses a horizontal distance of [latex]1800[/latex] meters at an angle of [latex]38°{.}[/latex] How long is the cable?
34.
Zelda is building the loft on her summer cottage. At its central point, the height of the loft is [latex]8[/latex] feet, and the pitch of the roof should be [latex]24°{.}[/latex] How long should the rafters be?
Exercise Group
For Problems 35–40, use a trig ratio to write an equation for [latex]x[/latex] in terms of [latex]\theta{.}[/latex]
35.
36.
37.
38.
39.
40.
Exercise Group
For Problems 41–44, find the altitude of the triangle. Round your answer to two decimal places.
41.
42.
43.
44.
Exercise Group
For Problems 45 and 46, find the length of the chord [latex]AB{.}[/latex] Round your answer to two decimal places.
45.
46.
Exercise Group
For Problems 47–50, fill in the table.
47.
| [latex]~~~~[/latex] |
sin |
cos |
tan |
| [latex]\theta[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
| [latex]\phi[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
48.
| [latex]~~~~[/latex] |
sin |
cos |
tan |
| [latex]\theta[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
| [latex]\phi[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
49.
| [latex]~~~~[/latex] |
sin |
cos |
tan |
| [latex]\theta[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
| [latex]\phi[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
50.
| [latex]~~~~[/latex] |
sin |
cos |
tan |
| [latex]\theta[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
| [latex]\phi[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
[latex]~~~~[/latex] |
51.
- In each of the figures for Problems 47-50, what is the relationship between the angles [latex]\theta[/latex] and [latex]\phi{?}[/latex]
- Study the tables for Problems 47-50. What do you notice about the values of sine and cosine for the angles [latex]\theta[/latex] and [latex]\phi{?}[/latex] Explain why this is true.
52.
There is a relationship between the tangent, the sine, and the cosine of any angle. Study the tables for Problems 47-50 to discover this relationship. Write your answer as an equation.
53.
- Use the figure to explain what happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases, and why.
- Use the figure to explain what happens to [latex]\cos \theta[/latex] as [latex]\theta[/latex] increases, and why.
54.
- Fill in the table for values of tan [latex]\theta{.}[/latex] Round your answers to four decimal places.
| [latex]\theta[/latex] |
[latex]~~0 °[/latex] |
[latex]~10 °[/latex] |
[latex]~20 °[/latex] |
[latex]~30 °[/latex] |
[latex]~40 °[/latex] |
[latex]~50 °[/latex] |
[latex]~60 °[/latex] |
[latex]~70 °[/latex] |
[latex]~80 °[/latex] |
| [latex]\tan \theta[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
- Fill in the table for values of tan [latex]\theta{.}[/latex] Round your answers to three decimal places.
| [latex]\theta[/latex] |
[latex]~81 °[/latex] |
[latex]~82 °[/latex] |
[latex]~83 °[/latex] |
[latex]~84 °[/latex] |
[latex]~85 °[/latex] |
[latex]~86 °[/latex] |
[latex]~87 °[/latex] |
[latex]~88 °[/latex] |
[latex]~89 °[/latex] |
| [latex]\tan \theta[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
- What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases?
- What value does your calculator give for [latex]\tan 90°{?}[/latex] Why?
55.
Explain why it makes sense that [latex]\sin 0° = 0[/latex] and [latex]\sin 90° = 1{.}[/latex] Use a figure to illustrate your explanation.
56.
Explain why it makes sense that [latex]\cos 0° = 1[/latex] and [latex]\cos 90° = 0{.}[/latex] Use a figure to illustrate your explanation
Exercise Group
For Problems 57–60, explain why the trigonometric ratio is not correct.
57.
[latex]\sin \theta = \dfrac{5}{9}[/latex]
58.
[latex]\tan \theta = \dfrac{4}{7}[/latex]
59.
[latex]\cos \theta = \dfrac{21}{20}[/latex]
60.
[latex]\sin \theta = \dfrac{8}{10}[/latex]
Exercise Group
For Problems 61–64, sketch and label a right triangle, then fill in the blank.
61.
If [latex]\sin \theta = 0.2358\text{,}[/latex] then [latex]\cos (90^{o} - \theta)=\underline{\qquad} \text{,}[/latex]
- If [latex]\cos \alpha = \dfrac{3}{11} \text{,}[/latex] then [latex]\underline{\qquad} (90° - \alpha) = \dfrac{3}{11}\text{.}[/latex]
- If [latex]\sin 42° = n\text{,}[/latex] then [latex]\cos \underline{\qquad} = n\text{.}[/latex]
- If [latex]\cos 13° = z\text{,}[/latex] then [latex]\sin \underline{\qquad} = z\text{.}[/latex]
62.
- If [latex]\cos \beta = \dfrac{2}{\sqrt{7}}{,}[/latex] then [latex]\sin (90° - \beta) =\underline{\qquad} {.}[/latex]
- If [latex]\sin \phi = 0.693{,}[/latex] then [latex](90° - \phi) = 0.693{.}[/latex]
- If [latex]\cos 87° = p{,}[/latex] then [latex]\sin \underline{\qquad} = p{.}[/latex]
- If [latex]\sin 59° =w{,}[/latex] then [latex]\cos \underline{\qquad} = w{.}[/latex]
63.
- If [latex]\sin \phi = \dfrac{5}{13}[/latex] and [latex]\cos \phi = \dfrac{12}{13}{,}[/latex] then [latex]\tan \phi =\underline{\qquad} {.}[/latex]
- If [latex]\cos \beta = \dfrac{1}{\sqrt{10}}{,}[/latex] and [latex]\sin \beta = \dfrac{3}{\sqrt{10}}{,}[/latex] then [latex]\tan \beta = \underline{\qquad}{.}[/latex]
- If [latex]\tan B = \dfrac{2}{\sqrt{5}}[/latex] and [latex]\cos B = \dfrac{\sqrt{5}}{3}{,}[/latex] then [latex]\sin B =\underline{\qquad}{.}[/latex]
- If [latex]\sin W = \sqrt{\dfrac{3}{7}}[/latex] and [latex]\tan W = \dfrac{\sqrt{3}}{2}{,}[/latex] then [latex]\cos W =\underline{\qquad}{.}[/latex]
64.
- If [latex]\cos \theta = \dfrac{2}{\sqrt{10}}[/latex] and [latex]\sin \theta = \sqrt{\dfrac{3}{5}}{,}[/latex] then [latex]\tan \theta = \underline{\qquad}{.}[/latex]
- If [latex]\sin \alpha = \dfrac{\sqrt{2}}{4}{,}[/latex] and [latex]\cos \alpha =\dfrac{\sqrt{14}}{4}{,}[/latex] then [latex]\tan \alpha = \underline{\qquad}{.}[/latex]
- If [latex]\tan A = \dfrac{\sqrt{7}}{3}[/latex] and [latex]\cos A = \dfrac{3}{4}{,}[/latex] then [latex]\sin A =\underline{\qquad}{.}[/latex]
- If [latex]\sin V = \sqrt{\dfrac{10}{5}}[/latex] and [latex]\tan V = \dfrac{2}{5}{,}[/latex] then [latex]\cos V =\underline{\qquad}{.}[/latex]
65.
Explain why the cosine of a [latex]73°[/latex] angle is always the same, no matter what size triangle the angle is in. Illustrate your explanation with a sketch.
66.
- Use your calculator to fill in a table of values for cos [latex]\theta{,}[/latex] rounded to hundredths.
| [latex]\theta[/latex] |
[latex]~~0 °[/latex] |
[latex]~15 °[/latex] |
[latex]~30 °[/latex] |
[latex]~45 °[/latex] |
[latex]~60 °[/latex] |
[latex]~75 °[/latex] |
[latex]~90 °[/latex] |
| [latex]\cos \theta[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
- If you plotted the points in your table, would they lie on a straight line? Why or why not?
67.
- What is the slope of the line through the origin and point [latex]P{?}[/latex]
- What is the tangent of the angle [latex]\theta{?}[/latex]
- On the same grid, sketch an angle whose tangent is [latex]\dfrac{8}{5}.[/latex]
68.
-
- Use your calculator to complete the table. Round your answers to hundredths.
| [latex]\theta[/latex] |
[latex]~14 °[/latex] |
[latex]~22 °[/latex] |
[latex]~35 °[/latex] |
[latex]~42 °[/latex] |
[latex]~58 °[/latex] |
[latex]~78 °[/latex] |
| [latex]\tan \theta[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
[latex]~~~[/latex] |
- Use the values of tan [latex]\theta[/latex] to sketch all the angles listed in the table. Locate the vertex of each angle at the origin and the initial side along the positive [latex]x[/latex]-axis.