Exercises: 2.2 Right Triangle Trigonometry

1.

Here are two right triangles with a [latex]65 °[/latex] angle.

1. Measure the sides [latex]AB[/latex] and [latex]BC[/latex] with a ruler. Use the lengths to estimate [latex]\sin 65°{.}[/latex]
2. Measure the sides [latex]AD[/latex] and [latex]DE[/latex] with a ruler. Use the lengths to estimate [latex]\sin 65°{.}[/latex]
3. 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.

1. Measure the sides [latex]AB[/latex] and [latex]AC[/latex] with a ruler. Use the lengths to estimate [latex]\cos 40°{.}[/latex]
2. Measure the sides [latex]AD[/latex] and [latex]AE[/latex] with a ruler. Use the lengths to estimate [latex]\cos 40°{.}[/latex]
3. 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]

5.

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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]

17.

18.

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21.

22.

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

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?

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48.

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51.

1. In each of the figures for Problems 47-50, what is the relationship between the angles [latex]\theta[/latex] and [latex]\phi{?}[/latex]
2. 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.

1. Use the figure to explain what happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases, and why.
2. Use the figure to explain what happens to [latex]\cos \theta[/latex] as [latex]\theta[/latex] increases, and why.

54.

1. 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]

2. 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]

3. What happens to [latex]\tan \theta[/latex] as [latex]\theta[/latex] increases?
4. 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

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]

61.

If [latex]\sin \theta = 0.2358\text{,}[/latex]  then [latex]\cos (90^{o} - \theta)=\underline{\qquad} \text{,}[/latex]

62.

1. If [latex]\cos \beta = \dfrac{2}{\sqrt{7}}{,}[/latex] then [latex]\sin (90° - \beta) =\underline{\qquad} {.}[/latex]
2. If [latex]\sin \phi = 0.693{,}[/latex] then [latex](90° - \phi) = 0.693{.}[/latex]
3. If [latex]\cos 87° = p{,}[/latex] then [latex]\sin \underline{\qquad} = p{.}[/latex]
4. If [latex]\sin 59° =w{,}[/latex] then [latex]\cos \underline{\qquad} = w{.}[/latex]

63.

1. If [latex]\sin \phi = \dfrac{5}{13}[/latex] and [latex]\cos \phi = \dfrac{12}{13}{,}[/latex] then [latex]\tan \phi =\underline{\qquad} {.}[/latex]
2. 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]
3. 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]
4. 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.

1. 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]
2. 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]
3. If [latex]\tan A = \dfrac{\sqrt{7}}{3}[/latex] and [latex]\cos A = \dfrac{3}{4}{,}[/latex] then [latex]\sin A =\underline{\qquad}{.}[/latex]
4. 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.

1. 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]

2. If you plotted the points in your table, would they lie on a straight line? Why or why not?

67.

1. What is the slope of the line through the origin and point [latex]P{?}[/latex]
2. What is the tangent of the angle [latex]\theta{?}[/latex]
3. On the same grid, sketch an angle whose tangent is [latex]\dfrac{8}{5}.[/latex]

68.

1. 1. Use your calculator to complete the table. Round your answers to hundredths.

1. 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.