Right Triangle Trigonometry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Right Triangle Trigonometry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The three primary trigonometric ratios—sine, cosine, and tangent—defined as ratios of specific sides in a right triangle.

Imagine a ramp leaning against a wall. The steepness depends on the ratio of how high the wall is to how long the ramp is. Trigonometry gives names to these ratios: sine is how high compared to the ramp, cosine is how far along the ground compared to the ramp, and tangent is how high compared to how far along the ground. No matter how big or small the ramp, if the angle is the same, these ratios stay the same.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Each acute angle in a right triangle determines fixed ratios between the sides, regardless of the triangle's size.

Common stuck point: Which side is 'opposite' and which is 'adjacent' depends on which angle you're looking at—they swap when you switch angles.

Worked Examples

Example 1

easy
In a right triangle, the angle \theta = 30°, and the hypotenuse is 10. Find the lengths of the opposite and adjacent sides.

Solution

  1. 1
    Step 1: Recall the definitions: \sin\theta = \frac{\text{opp}}{\text{hyp}} and \cos\theta = \frac{\text{adj}}{\text{hyp}}.
  2. 2
    Step 2: Find the opposite side: \sin 30° = \frac{\text{opp}}{10}. Since \sin 30° = 0.5, we get \text{opp} = 10 \times 0.5 = 5.
  3. 3
    Step 3: Find the adjacent side: \cos 30° = \frac{\text{adj}}{10}. Since \cos 30° = \frac{\sqrt{3}}{2} \approx 0.866, we get \text{adj} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66.

Answer

Opposite = 5, Adjacent = 5\sqrt{3}.
The sine ratio connects the opposite side to the hypotenuse, and the cosine ratio connects the adjacent side to the hypotenuse. For a 30° angle, these are well-known values: \sin 30° = \frac{1}{2} and \cos 30° = \frac{\sqrt{3}}{2}. Multiplying each by the hypotenuse length gives the side lengths.

Example 2

medium
A ladder 13 feet long leans against a wall. The base of the ladder is 5 feet from the wall. Find the angle the ladder makes with the ground (to the nearest degree).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
In a right triangle with angle \theta = 45° and hypotenuse = 8, find both legs.

Example 2

hard
From the top of a 50-meter tall lighthouse, the angle of depression to a boat is 32°. How far is the boat from the base of the lighthouse (to the nearest meter)?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

trianglespythagorean theoremratios