Applications of Right Triangle Trigonometry

What You’ll Learn

In this lesson you’ll use sine, cosine, and tangent to solve practical problems involving heights, distances, and angles in right triangles.

Right triangle trigonometry is extremely useful because many real-world situations can be modeled with right triangles. The key is to:

Draw a clear diagram and label the known and unknown parts.
Identify which side is opposite, adjacent, or the hypotenuse relative to the given angle.
Choose the correct ratio (SOH-CAH-TOA).
Set up the equation and solve.
Check that your answer makes sense in context.

Two terms you’ll see a lot in application problems:

Angle of elevation: the angle measured upward from horizontal (looking up at something)
Angle of depression: the angle measured downward from horizontal (looking down at something)

Both create right triangles you can solve with trig.

Example 1: A 25-foot ladder leans against a building and makes a 68° angle with the ground. How high up the building does the ladder reach?

The height is the side opposite the 68° angle, and the ladder is the hypotenuse. That’s SOH: sin = opposite / hypotenuse.

\sin(68°) = \frac{\text{height}}{25}

\text{Height} = 25 \times \sin(68°) \approx 25 \times 0.9272 \approx 23.18 \text{ feet}

Example 2: From a point 120 feet away from the base of a tree, the angle of elevation to the top is 52°. How tall is the tree?

The tree height is the opposite side and the 120 feet is the adjacent side. That’s TOA: tan = opposite / adjacent.

\tan(52°) = \frac{\text{height}}{120}

\text{Height} = 120 \times \tan(52°) \approx 120 \times 1.2799 \approx 153.59 \text{ feet}

Example 3: A ramp must rise 4 feet over a horizontal distance of 30 feet. What angle does the ramp make with the ground?

Here we know the opposite (4 ft) and adjacent (30 ft) and need the angle. That’s TOA again, but solved backwards with inverse tangent.

\tan(\theta) = \frac{4}{30} \approx 0.1333

\theta = \tan^{-1}(0.1333) \approx 7.6°

Right triangle trigonometry is used daily in:

Construction (calculating roof slopes, ramp grades, stair angles)
Surveying and forestry (measuring tree heights or distances across obstacles)
Safety engineering (proper ladder placement, ideally around 75° with the ground)
Navigation and hiking (estimating heights or distances)
Architecture and design (creating accessible ramps that meet ADA standards)

Example: building codes often require wheelchair ramps to have a maximum slope of 1:12 (about 4.76°). Trigonometry helps verify compliance.

A 30-foot ladder makes a 70° angle with the ground. How high up the wall does it reach?

From 80 feet away, the angle of elevation to the top of a building is 48°. How tall is the building?

A ramp rises 3 feet over a horizontal distance of 36 feet. What is the angle?

The most important first step in a trig application problem is:

An angle of elevation is measured: