What if a restaurant needed to build a wheelchair ramp for its customers? The angle of elevation for a ramp is recommended to be \begin{align*}5^\circ\end{align*}. If the vertical distance from the sidewalk to the front door is two feet, what is the horizontal distance that the ramp will take up \begin{align*}(x)\end{align*}? How long will the ramp be \begin{align*}(y)\end{align*}? Round your answers to the nearest hundredth.

### Trigonometry Word Problems

A practical application of the trigonometric functions is to find the measure of lengths that you cannot measure. Very frequently, angles of depression and elevation are used in these types of problems.

**Angle of Depression:** The angle measured from the horizon or horizontal line, down.

**Angle of Elevation:** The angle measure from the horizon or horizontal line, up.

#### The Washington Monument

An inquisitive math student is standing 25 feet from the base of the Washington Monument. The angle of elevation from her horizontal line of sight is \begin{align*}87.4^\circ\end{align*}. If her “eye height” is 5ft, how tall is the monument?

**Solution:** We can find the height of the monument by using the tangent ratio and then adding the eye height of the student.

\begin{align*}\tan 87.4^\circ & = \frac{h}{25}\\ h & = 25 \cdot \tan 87.4^\circ = 550.54\end{align*}

Adding 5 ft, the total height of the Washington Monument is 555.54 ft.

According to Wikipedia, the actual height of the monument is 555.427 ft.

#### Solving for an Angle Measurment

A 25 foot tall flagpole casts a 42 foot shadow. What is the angle that the sun hits the flagpole?

Draw a picture. The angle that the sun hits the flagpole is \begin{align*}x^\circ\end{align*}. We need to use the inverse tangent ratio.

\begin{align*}\tan x &= \frac{42}{25}\\ \tan^{-1} \frac{42}{25} & \approx 59.2^\circ = x\end{align*}

#### Solving for the Angle of Depression

Elise is standing on top of a 50 foot building and sees her friend, Molly. If Molly is 30 feet away from the base of the building, what is the angle of depression from Elise to Molly? Elise’s eye height is 4.5 feet.

Because of parallel lines, the angle of depression is equal to the angle at Molly, or \begin{align*}x^\circ\end{align*}. We can use the inverse tangent ratio.

\begin{align*}\tan^{-1} \left( \frac{54.5}{30} \right) = 61.2^\circ = x\end{align*}

#### Ramp Problem Revisited

To find the horizontal length and the actual length of the ramp, we need to use the tangent and sine.

\begin{align*}\tan 5^\circ & = \frac{2}{x} && \sin 5^\circ = \frac{2}{y}\\ x & = \frac{2}{\tan 5^\circ} = 22.86 && \qquad y = \frac{2}{\sin 5^\circ} = 22.95\end{align*}

### Examples

#### Example 1

Mark is flying a kite and realizes that 300 feet of string are out. The angle of the string with the ground is \begin{align*}42.5^\circ\end{align*}. How high is Mark's kite above the ground?

It might help to draw a picture. Then write and solve a trig equation.

\begin{align*} \sin 42.5^\circ &=\frac{x}{300} \\ 300 \cdot \sin 42.5^\circ &=x \\ x & \approx 202.7\end{align*}

The kite is about \begin{align*}202.7\end{align*} feet off of the ground.

#### Example 2

A 20 foot ladder rests against a wall. The base of the ladder is 7 feet from the wall. What angle does the ladder make with the ground?

It might help to draw a picture.

\begin{align*}\cos x &=\frac{7}{20} \\ x&=\cos^{-1}\frac{7}{20}\\ x & \approx 69.5^\circ\end{align*}

#### Example 3

A 20 foot ladder rests against a wall. The ladder makes a \begin{align*}55^\circ\end{align*} angle with the ground. How far from the wall is the base of the ladder?

It might help to draw a picture.

\begin{align*}\cos 55^\circ &=\frac{x}{20} \\ 20\cdot \cos 55^\circ &=x \\ x & \approx 11.5 ft\end{align*}.

### Review

- Kristin is swimming in the ocean and notices a coral reef below her. The angle of depression is \begin{align*}35^\circ\end{align*} and the depth of the ocean, at that point is 250 feet. How far away is she from the reef?
- The Leaning Tower of Pisa currently “leans” at a \begin{align*}4^\circ\end{align*} angle and has a vertical height of 55.86 meters. How tall was the tower when it was originally built?
- The angle of depression from the top of an apartment building to the base of a fountain in a nearby park is \begin{align*}72^\circ\end{align*}. If the building is 78 ft tall, how far away is the fountain?
- William spots a tree directly across the river from where he is standing. He then walks 20 ft upstream and determines that the angle between his previous position and the tree on the other side of the river is \begin{align*}65^\circ\end{align*}. How wide is the river?
- Diego is flying his kite one afternoon and notices that he has let out the entire 120 ft of string. The angle his string makes with the ground is \begin{align*}52^\circ\end{align*}. How high is his kite at this time?
- A tree struck by lightning in a storm breaks and falls over to form a triangle with the ground. The tip of the tree makes a \begin{align*}36^\circ\end{align*} angle with the ground 25 ft from the base of the tree. What was the height of the tree to the nearest foot?
- Upon descent an airplane is 20,000 ft above the ground. The air traffic control tower is 200 ft tall. It is determined that the angle of elevation from the top of the tower to the plane is \begin{align*}15^\circ\end{align*}. To the nearest mile, find the ground distance from the airplane to the tower.
- A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building?
- Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation?
- A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck?
- Elizabeth wants to know the angle at which the sun hits a tree in her backyard at 3 pm. She finds that the length of the tree’s shadow is 24 ft at 3 pm. At the same time of day, her shadow is 6 ft 5 inches. If Elizabeth is 4 ft 8 inches tall, find the height of the tree and hence the angle at which the sunlight hits the tree.
- Alayna is trying to determine the angle at which to aim her sprinkler nozzle to water the top of a 5 ft bush in her yard. Assuming the water takes a straight path and the sprinkler is on the ground 4 ft from the tree, at what angle of inclination should she set it?
- Tommy was solving the triangle below and made a mistake. What did he do wrong? \begin{align*}\tan^{-1} \left ( \frac{21}{28} \right ) \approx 36.9^\circ\end{align*}
- Tommy then continued the problem and set up the equation: \begin{align*}\cos 36.9^\circ = \frac{21}{h}\end{align*}. By solving this equation he found that the hypotenuse was 26.3 units. Did he use the correct trigonometric ratio here? Is his answer correct? Why or why not?
- How could Tommy have found the hypotenuse in the triangle another way and avoided making his mistake?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.8.