Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

Angles of Elevation and Depression

• The angle of elevation is the angle measured upward from the horizontal to the line of sight toward an object above.
• The angle of depression is the angle measured downward from the horizontal to the line of sight toward an object below.
• These two angles are equal when measured from two ends of the same line of sight, because they are alternate interior angles between parallel horizontal lines.

Right-Angled Trigonometry (SOH CAH TOA)

For a right triangle with angle $\theta$:

Inverse Trigonometric Functions

When side lengths are known and an angle must be found, use the inverse functions:

$\theta ={\sin }^{-1}\left(\frac{\text{Opp}}{\text{Hyp}}\right),\theta ={\cos }^{-1}\left(\frac{\text{Adj}}{\text{Hyp}}\right),\theta ={\tan }^{-1}\left(\frac{\text{Opp}}{\text{Adj}}\right)$

Proportional Reasoning with Shadows

At the same time of day, the sun's angle of elevation $\theta$ is constant. If a flagpole of height $h_1$ casts a shadow $s_1$, and a building casts a shadow $s_2$, then:

$\frac{h_1}{s_1}=\tan (\theta )=\frac{h_2}{s_2}⟹h_2=\frac{h_1\cdot s_2}{s_1}$

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Worked Examples

Example 1 — Finding Height Using Angle of Elevation

Problem: A person stands 50 m from the base of a building and measures the angle of elevation to the top as $32°$. Find the height of the building.

Solution: $\tan (32°)=\frac{h}{50}⟹h=50\times \tan (32°)\approx 50\times 0.6249\approx 31.2\text{ m}$

Example 2 — Finding an Angle Using Inverse Trig

Problem: A ramp rises 1.2 m over a horizontal distance of 6 m. Find the angle of inclination.

Solution: $\tan (\theta )=\frac{1.2}{6}=0.2⟹\theta ={\tan }^{-1}(0.2)\approx 11.3°$

Example 3 — Angle of Depression

Problem: From the top of a 40 m cliff, the angle of depression to a boat is $25°$. How far is the boat from the base of the cliff?

Solution: The angle of depression equals the angle of elevation from the boat, so: $\tan (25°)=\frac{40}{d}⟹d=\frac{40}{\tan (25°)}\approx \frac{40}{0.4663}\approx 85.8\text{ m}$

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Summary

This exercise applies basic trigonometric principles to practical engineering, architecture, and surveying problems. The primary strategy involves:

1. Identifying the right-angled triangle within the word problem.
2. Labelling the known components (angles or sides).
3. Selecting the appropriate trigonometric ratio (SOH CAH TOA) to solve for the unknown.
4. Using inverse trig functions (${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$) when an angle must be found from side lengths.
5. Applying similar-triangle proportions for shadow problems.