2.2 Right Triangle Trigonometry

Find the sine of the labeled angle in the triangle at right. Round your answer to 4 decimal places.

1. Use your calculator to complete the table, rounding your answers to four decimal places.

   $\theta$	$0^{\circ }$	${10}^{\circ }$	${20}^{\circ }$	${30}^{\circ }$	${40}^{\circ }$	${50}^{\circ }$	${60}^{\circ }$	${70}^{\circ }$	${80}^{\circ }$	${90}^{\circ }$
   $\sin (\theta )$	$$	$$	$$	$$	$$	$$	$$	$$	$$	$$

2. What do you notice about the values of $\sin (\theta )$ as $\theta$ increases from $0^{\circ }$ to ${90}^{\circ }$? If you plot the values of $\sin (\theta )$ against the values of $\theta$, will the graph be a straight line? Why or why not?

Find the length of the hypotenuse in the triangle shown.

1. Use the tangent ratio to find $x$ in the triangle shown.

   

2. Use the sine ratio to find the hypotenuse, $c$, of the triangle.
3. Use the Pythagorean theorem to find the hypotenuse of the triangle. Do you get the same answer with both methods? Can you explain why the calculations might give (slightly) different answers?

Another isosceles triangle has base angles of ${72}^{\circ }$ and equal sides of length 6.8 centimeters. Find the length of the base.

Here are two right triangles with a ${65}^{\circ }$ angle.

1. Measure the sides $AB$ and $BC$ with a ruler. Use the lengths to estimate $\sin ({65}^{\circ })$.
2. Measure the sides $AD$ and $DE$ with a ruler. Use the lengths to estimate $\sin ({65}^{\circ })$.
3. Use your calculator to look up $\sin ({65}^{\circ })$. Compare your answers. How close were your estimates?

Use the figure in Problem 1 to calculate two estimates each for the cosine and tangent of ${65}^{\circ }$. Compare your estimates to your calculator's values for $\cos ({65}^{\circ })$ and $\tan ({65}^{\circ })$.

Here are two right triangles with a ${40}^{\circ }$ angle.

1. Measure the sides $AB$ and $AC$ with a ruler. Use the lengths to estimate $\cos ({40}^{\circ })$.
2. Measure the sides $AD$ and $AE$ with a ruler. Use the lengths to estimate $\cos ({40}^{\circ })$.
3. Use your calculator to look up $\cos ({40}^{\circ })$. Compare your answers. How close were your estimates?

Use the figure in Problem 2 to calculate two estimates each for the cosine and tangent of ${40}^{\circ }$. Compare your estimates to your calculator's values for $\sin ({40}^{\circ })$ and $\tan ({40}^{\circ })$.

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$$\cos (\theta )={\displaystyle \frac{3}{5}}$$

$$\tan (\theta )={\displaystyle \frac{7}{2}}$$

$$\tan (\theta )={\displaystyle \frac{11}{4}}$$

$$\sin (\theta )={\displaystyle \frac{4}{9}}$$

$$\sin (\theta )={\displaystyle \frac{1}{9}}$$

$$\cos (\theta )={\displaystyle \frac{7}{8}}$$

One angle is ${40}^{\circ }$, the side opposite that angle is 8 inches

One angle is ${65}^{\circ }$, the side adjacent to that angle is 30 yards

One angle is ${28}^{\circ }$, the hypotenuse is 56 feet

One leg is 15 meters, the hypotenuse is 18 meters

To measure the height of cloud cover, airport controllers fix a searchlight to shine a vertical beam on the clouds. The searchlight is 120 yards from the office. A technician in the office measures the angle of elevation to the light on the cloud cover at ${54.8}^{\circ }$. What is the height of the cloud cover?

To measure the distance across a canyon, Evel first sights an interesting rock directly opposite on the other side. He then walks 200 yards down the rim of the canyon and sights the rock again, this time at an angle of ${18.5}^{\circ }$ from the canyon rim. What is the width of the canyon?

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 ${36.2}^{\circ }$. The depth of the ocean at their location is 260 feet. How far should they move so that they are directly above the wrecked vessel?

Ramps for wheelchairs should be no steeper than an angle of $6^{\circ }$. How much horizontal distance should be allowed for a ramp that rises 5 feet in height?

The radio signal from a weather balloon indicates that it is 1500 meters from a meteorologist on the ground. The angle of elevation to the balloon is ${48}^{\circ }$. What is the balloon's altitude?

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 ${27}^{\circ }$ and the string to the kite was 1850 feet long. How far did Han Xin's troops have to dig?

A cable car on a ski lift traverses a horizontal distance of 1800 meters at an angle of ${38}^{\circ }$. How long is the cable?

Zelda is building the loft on her summer cottage. At its central point, the height of the loft is 8 feet, and the pitch of the roof should be ${24}^{\circ }$. How long should the rafters be?

1. In each of the figures for Problems 47-50, what is the relationship between the angles $\theta$ and $\varphi$?
2. Study the tables for Problems 47-50. What do you notice about the values of sine and cosine for the angles $\theta$ and $\varphi$? Explain why this is true.

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.

1. Use the figure to explain what happens to $\tan (\theta )$ as $\theta$ increases, and why.
2. Use the figure to explain what happens to $\cos (\theta )$ as $\theta$ increases, and why.

1. Fill in the table for values of $\tan (\theta )$. Round your answers to four decimal places.

   $\theta$	$0^{\circ }$	${10}^{\circ }$	${20}^{\circ }$	${30}^{\circ }$	${40}^{\circ }$	${50}^{\circ }$	${60}^{\circ }$	${70}^{\circ }$	${80}^{\circ }$
   $\tan (\theta )$	$$	$$	$$	$$	$$	$$	$$	$$	$$

2. Fill in the table for values of $\tan (\theta )$. Round your answers to three decimal places.

   $\theta$	${81}^{\circ }$	${82}^{\circ }$	${83}^{\circ }$	${84}^{\circ }$	${85}^{\circ }$	${86}^{\circ }$	${87}^{\circ }$	${88}^{\circ }$	${89}^{\circ }$
   $\tan (\theta )$	$$	$$	$$	$$	$$	$$	$$	$$	$$

3. What happens to $\tan (\theta )$ as $\theta$ increases?
4. What value does your calculator give for $\tan ({90}^{\circ })$? Why?

Explain why it makes sense that $\sin (0^{\circ })=0$ and $\sin ({90}^{\circ })=1$. Use a figure to illustrate your explanation.

Explain why it makes sense that $\cos (0^{\circ })=1$ and $\cos ({90}^{\circ })=0$. Use a figure to illustrate your explanation

$$\sin (\theta )={\displaystyle \frac{5}{9}}$$

$$\tan (\theta )={\displaystyle \frac{4}{7}}$$

$$\cos (\theta )={\displaystyle \frac{21}{20}}$$

$$\sin (\theta )={\displaystyle \frac{8}{10}}$$

1. If $\sin (\theta )=0.2358$, then $\cos ({90}^{\circ }-\theta )=\underset{―}{}$.
2. If $\cos (\alpha )={\displaystyle \frac{3}{11}}$, then $\underset{―}{}({90}^{\circ }-\alpha )={\displaystyle \frac{3}{11}}$.
3. If $\sin ({42}^{\circ })=n$, then $\cos (\underset{―}{})=n$.
4. If $\cos ({13}^{\circ })=z$, then $\sin (\underset{―}{})=z$.

1. If $\cos (\beta )={\displaystyle \frac{2}{\sqrt{7}}}$, then $\sin ({90}^{\circ }-\beta )=\underset{―}{}$.
2. If $\sin (\varphi )=0.693$, then ______$({90}^{\circ }-\varphi )=0.693$.
3. If $\cos ({87}^{\circ })=p$, then $\sin (\underset{―}{})=p$.
4. If $\sin ({59}^{\circ })=w$, then $\cos (\underset{―}{})=w$.

1. If $\sin (\varphi )={\displaystyle \frac{5}{13}}$ and $\cos (\varphi )={\displaystyle \frac{12}{13}}$, then $\tan (\varphi )=\underset{―}{}$.
2. If $\cos (\beta )={\displaystyle \frac{1}{\sqrt{10}}}$, and $\sin (\beta )={\displaystyle \frac{3}{\sqrt{10}}}$, then $\tan (\beta )=\underset{―}{}$.
3. If $\tan (B)={\displaystyle \frac{2}{\sqrt{5}}}$ and $\cos (B)={\displaystyle \frac{\sqrt{5}}{3}}$, then $\sin (B)=\underset{―}{}$.
4. If $\sin (W)=\sqrt{{\displaystyle \frac{3}{7}}}$ and $\tan (W)={\displaystyle \frac{\sqrt{3}}{2}}$, then $\cos (W)=\underset{―}{}$.

1. If $\cos (\theta )={\displaystyle \frac{2}{\sqrt{10}}}$ and $\sin (\theta )=\sqrt{{\displaystyle \frac{3}{5}}}$, then $\tan (\theta )=\underset{―}{}$.
2. If $\sin (\alpha )={\displaystyle \frac{\sqrt{2}}{4}}$, and $\cos (\alpha )={\displaystyle \frac{\sqrt{14}}{4}}$, then $\tan (\alpha )=\underset{―}{}$.
3. If $\tan (A)={\displaystyle \frac{\sqrt{7}}{3}}$ and $\cos (A)={\displaystyle \frac{3}{4}}$, then $\sin (A)=\underset{―}{}$.
4. If $\sin (V)={\displaystyle \frac{\sqrt{10}}{5}}$ and $\tan (V)=\sqrt{{\displaystyle \frac{2}{3}}}$, then $\cos (V)=\underset{―}{}$.

Explain why the cosine of a ${73}^{\circ }$ angle is always the same, no matter what size triangle the angle is in. Illustrate your explanation with a sketch.

1. Use your calculator to fill in a table of values for $\cos (\theta )$, rounded to hundredths.

   $\theta$	$0^{\circ }$	${15}^{\circ }$	${30}^{\circ }$	${45}^{\circ }$	${60}^{\circ }$	${75}^{\circ }$	${90}^{\circ }$
   $\cos (\theta )$	$$	$$	$$	$$	$$	$$	$$

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

1. What is the slope of the line through the origin and point $P$?
2. What is the tangent of the angle $\theta$?
3. On the same grid, sketch an angle whose tangent is ${\displaystyle \frac{8}{5}}.$

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

   $\theta$	${14}^{\circ }$	${22}^{\circ }$	${35}^{\circ }$	${42}^{\circ }$	${58}^{\circ }$	${78}^{\circ }$
   $\tan (\theta )$	$$	$$	$$	$$	$$	$$

2. Use the values of $\tan (\theta )$ 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 $x$-axis.