5.3: Applications of Trigonometry
- Page ID
- 34144
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
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.
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.
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?
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
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.
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)?
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?
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)?