5.3: Applications of Trigonometry

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Trigonometry has many applications in science and engineering. In this section we will present just a few examples from surveying and navigation.

The angle made by the line of sight of an observer on the ground to a point above the horizontal is called the angle of elevation. In Figure $\PageIndex{1}$, $\angle BAC$ is the angle of elevation.

Figure $\PageIndex{1}$: The angle of elevation.

Example $\PageIndex{1}$

At a point 50 feet from a tree the angle of elevation of the top of the tree is $43^{\circ}$. Find the height of the tree to the nearest tenth of a foot.

$clipboard_eb172ae95cdf1e7bc1541573428478171.png$

Solution

Let $x=$ height of tree.

$\begin{array} {rcl} {\tan 43^{\circ}} & = & {\dfrac{x}{50}} \\ {.9325} & = & {\dfrac{x}{50}} \\ {(50)(.9325)} & = & {\dfrac{x}{50}(50)} \\ {46.6250} & = & {x} \\ {46.6} & = & {x} \end{array}$

Answer: $x = 46.6$ feet.

The angle made by the line of sight of an observer above to a point on the ground is called the angle of depression. In FIgure $\PageIndex{2}$, $\angle ABD$ is the angle of depression.

Figure $\PageIndex{2}$: The angle of depression

Example $\PageIndex{2}$

From an airplane 5000 feet above the ground the angle of depression of an airport is $5^{\circ}$. How far away is the airport to the nearest hundred feet?

$屏幕快照 2020-11-24 下午1.05.54.png$

Solution

Let $x =$ distance to airport. $\angle ABC = 85^{\circ}$.

$\begin{array} {rcl} {\cos 85^{\circ}} & = & {\dfrac{5000}{x}} \\ {.0872} & = & {\dfrac{5000}{x}} \\ {.0872x} & = & {5000} \\ {x} & = & {\dfrac{5000}{.0872} = 57,300} \end{array}$

Answer: 57,300 feet

Example $\PageIndex{3}$

A road rises 30 feet in a horizontal distance of 300 feet, Find to the nearest degree the angle the road makes with the horizontal.

$屏幕快照 2020-11-24 下午1.09.19.png$

Solution

$\begin{array} {rcl} {\tan A} & = & {\dfrac{30}{300}} \\ {\tan A} & = & {.1000} \\ {\angle A} & = & {6^{\circ}} \end{array}$

Answer: $6^{\circ}$.

Problems

1. At a point 60 feet from a tree the angle of elevation of the top of the tree is $40^{\circ}$. Find the height of the tree to the nearest tenth of a foot.

2. At a point 100 feet from a tall building the angle of elevation of the top of the building is $65^{\circ}$. Find the height of the building to the nearest foot.

3. From a helicopter 1000 feet above the ground the angle of depression of a helinort is $10^{\circ}$. How far away is the heliport to the nearest foot?

4. From the top of a 100 foot lighthouse the angle of depression of a boat is $15^{\circ}$. How far is the boat from the bottom of the lighthouse (nearest foot)?

$Screen Shot 2020-11-24 at 1.14.51 PM.png$

5. A road rises 10 feet in a horizontal distance of 400 feet. Find to the nearest degree the angle the road makes with the horizontal.

6. If a 20 foot telephone pole casts a shadow of 43 feet , what is the angle of elevation of the sun?

$Screen Shot 2020-11-24 at 1.15.17 PM.png$

7. A 20 foot ladder is leaning against a wall, It makes an angle of $70^{\circ}$ with the ground. How high is the top of the ladder from the ground (nearest tenth of a foot)?

8. The angle of elevation of the top of a mountain from a point 20 miles away is $6^{\circ}$. How high is the mountain (nearest tenth of a mile)?

Example \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Problems