Trigonometric word problems usually ask for a height, distance, or length that cannot be measured directly.
The main skill is to translate the situation into a triangle.
1. Right-triangle ratios
For a right triangle with angle \(\theta\):
\[
\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}
\]
\[
\tan\theta=\frac{\text{opposite}}{\text{adjacent}}
\]
Many real-life problems use tangent because they involve a vertical height and a horizontal distance.
2. Angle of elevation
An angle of elevation is measured upward from a horizontal line. For example, if an observer is \(d\)
meters from a building and the angle to the top is \(\theta\), then:
\[
\tan\theta=\frac{\text{vertical rise}}{d}
\]
\[
\text{vertical rise}=d\tan\theta
\]
If the observer’s eye height is \(e\), the total height is:
\[
H=e+d\tan\theta
\]
3. Angle of depression
An angle of depression is measured downward from a horizontal line. It often gives:
\[
x=\frac{\text{vertical drop}}{\tan\theta}
\]
This is useful for finding horizontal distance from a cliff, tower, aircraft, or observation point.
4. Ladder problems
If a ladder of length \(L\) makes an angle \(\theta\) with the ground, then:
\[
\text{height reached}=L\sin\theta
\]
\[
\text{distance from wall}=L\cos\theta
\]
5. Shadow problems
If an object casts a shadow of length \(s\) and the sun’s angle of elevation is \(\theta\), then:
\[
h=s\tan\theta
\]
6. Distance across a river
Sometimes the triangle is not a simple right triangle at first. If two observation points are
separated by a baseline \(L\), and the sight angles to the target are \(\alpha\) and \(\beta\), then
the perpendicular distance \(h\) to the target is:
\[
h=\frac{L\tan\alpha\tan\beta}{\tan\alpha+\tan\beta}
\]
7. Formula summary
| Situation |
Common formula |
Unknown found |
| Height from angle of elevation |
\(H=e+d\tan\theta\) |
Object height |
| Angle of depression |
\(x=\dfrac{H-h_t}{\tan\theta}\) |
Horizontal distance |
| Ladder |
\(h=L\sin\theta\) |
Height on wall |
| Shadow |
\(h=s\tan\theta\) |
Object height |
| River width |
\(h=\dfrac{L\tan\alpha\tan\beta}{\tan\alpha+\tan\beta}\) |
Perpendicular distance |
8. Common mistakes
- Do not confuse angle of elevation with angle of depression.
- Check whether the known side is opposite, adjacent, or hypotenuse.
- Use tangent for most height-and-distance problems.
- Use sine or cosine when the hypotenuse is known, such as in ladder problems.
- Keep angle units consistent: degrees with degrees, radians with radians.
- Draw a diagram before choosing the formula.
