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: Right-triangle trig turns an angle into a fixed ratio of two specific sides.

Common stuck point: The procedure for right triangle trigonometry is the easy part; the trap is mislabeling opposite and adjacent. Asking "Is there a right angle and an acute angle linking a pair of sides I need to relate?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there a right angle and an acute angle linking a pair of sides I need to relate?

Worked Examples

Example 1

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

Answer

Opposite =5= 5, Adjacent =53= 5\sqrt{3}.

First step

1
Step 1: Recall the definitions: sin⁡θ=opphyp\sin\theta = \frac{\text{opp}}{\text{hyp}} and cos⁡θ=adjhyp\cos\theta = \frac{\text{adj}}{\text{hyp}}.

Full solution

  1. 2
    Step 2: Find the opposite side: sin⁡30°=opp10\sin 30° = \frac{\text{opp}}{10}. Since sin⁡30°=0.5\sin 30° = 0.5, we get opp=10×0.5=5\text{opp} = 10 \times 0.5 = 5.
  2. 3
    Step 3: Find the adjacent side: cos⁡30°=adj10\cos 30° = \frac{\text{adj}}{10}. Since cos⁡30°=32≈0.866\cos 30° = \frac{\sqrt{3}}{2} \approx 0.866, we get adj=10×32=53≈8.66\text{adj} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66.
The sine ratio connects the opposite side to the hypotenuse, and the cosine ratio connects the adjacent side to the hypotenuse. For a 30°30° angle, these are well-known values: sin⁡30°=12\sin 30° = \frac{1}{2} and cos⁡30°=32\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).

Example 3

medium
A 24-foot ladder leans against a wall at an angle of 75°75° to the ground. How high up the wall does it reach (to the nearest tenth of a foot)?

Example 4

medium
From the ground, the angle of elevation to the top of a tree is 35°35°. You are 4040 m from the base. How tall is the tree (to the nearest tenth of a meter)?

Example 5

medium
A kite is at the end of 8080 m of string making an angle of 50°50° with the ground. How high is the kite (assume taut string, to the nearest tenth of a meter)?

Example 6

medium
Standing 2525 m from a building, you measure the angle of elevation to the top as 40°40°. Your eye is 1.61.6 m above the ground. How tall is the building (to the nearest tenth)?

Example 7

hard
From the top of a 6060-m cliff, the angle of depression to a boat is 20°20°. Two minutes later, it is 35°35°. How far did the boat travel (to the nearest meter)?

Example 8

hard
A regular hexagon has side length 66. Find the length of a diagonal connecting two vertices that are two apart (skipping one vertex), using right-triangle trigonometry.

Example 9

challenge
A right triangle has perimeter 3030 and one acute angle θ\theta such that sin⁡θ=3/5\sin\theta = 3/5. Find the lengths of all three sides.

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 θ=45°\theta = 45° and hypotenuse =8= 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°32°. How far is the boat from the base of the lighthouse (to the nearest meter)?

Example 3

easy
In a right triangle, sin⁡θ\sin\theta is the ratio of which two sides?

Example 4

easy
cos⁡θ\cos\theta is the ratio of which two sides?

Example 5

easy
tan⁡θ\tan\theta is the ratio of which two sides?

Example 6

easy
A right triangle has opposite 3 and hypotenuse 6 for angle θ\theta. Find sin⁡θ\sin\theta.

Example 7

easy
A right triangle has adjacent 4 and hypotenuse 5 for angle θ\theta. Find cos⁡θ\cos\theta.

Example 8

easy
A right triangle has opposite 6 and adjacent 8 for angle θ\theta. Find tan⁡θ\tan\theta.

Example 9

easy
What does the mnemonic SOH-CAH-TOA help you remember?

Example 10

easy
For angle θ\theta in a right triangle, which side is the 'opposite' side?

Example 11

medium
A ramp rises at angle θ\theta where the height is 3 and the ramp length (hypotenuse) is 6. Find θ\theta.

Example 12

medium
In a right triangle, angle θ=30°\theta = 30° and the hypotenuse is 10. Find the opposite side.

Example 13

medium
Why is mixing up the opposite and adjacent sides the most common trig error?

Example 14

medium
A right triangle has legs 5 and 12. Find sin⁡θ\sin\theta for the angle opposite the side of length 5.

Example 15

medium
A 20-ft ladder leans against a wall at 60° to the ground. How high up the wall does it reach?

Example 16

medium
A right triangle has tan⁡θ=1\tan\theta = 1. What is θ\theta?

Example 17

medium
In a right triangle, the two acute angles are θ\theta and 90°−θ90° - \theta. Why is sin⁡θ=cos⁡(90°−θ)\sin\theta = \cos(90° - \theta)?

Example 18

medium
From 50 m away, the angle of elevation to a tower top is 40°. Set up the equation for the tower's height hh.

Example 19

challenge
A right triangle has sin⁡θ=35\sin\theta = \frac{3}{5}. Find cos⁡θ\cos\theta and tan⁡θ\tan\theta without finding θ\theta.

Example 20

challenge
Two buildings are 30 m apart. From the top of the shorter (height 20 m), the angle of elevation to the top of the taller is 25°. Find the taller building's height.

Example 21

challenge
Prove that sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 using a right triangle.

Example 22

challenge
Why does the slope of a line equal the tangent of its angle of inclination?

Example 23

easy
In a right triangle, sin⁡θ=0.6\sin\theta = 0.6 and the hypotenuse is 1010. Find the opposite side.

Example 24

easy
In a right triangle, θ=60°\theta = 60° and the adjacent side is 55. Find the opposite side.

Example 25

easy
In a right triangle, cos⁡θ=0.8\cos\theta = 0.8 and the hypotenuse is 1515. Find the adjacent side.

Example 26

easy
A right triangle has legs 77 and 2424. Find the hypotenuse and sin⁡θ\sin\theta where θ\theta is opposite the leg of length 77.

Example 27

medium
A right triangle has hypotenuse 2020 and one angle θ=25°\theta = 25°. Find both legs (to the nearest tenth).

Example 28

medium
A ramp rises 1.51.5 m over a horizontal run of 99 m. What angle does it make with the ground (to the nearest tenth of a degree)?

Example 29

medium
A right triangle has legs a=9a = 9 and b=12b = 12. Find the two acute angles (to the nearest tenth of a degree).

Example 30

medium
Find the area of a right triangle whose hypotenuse is 2626 and one acute angle is 30°30°.

Example 31

medium
A right triangle has sin⁡θ=0.28\sin\theta = 0.28. Find cos⁡θ\cos\theta exactly (assume θ\theta acute).

Example 32

hard
From two points AA and BB, 3030 m apart on level ground, the angles of elevation to the top of a tower (directly behind BB from AA) are 30°30° and 45°45° respectively. Find the tower's height (exact form).

Example 33

hard
A right triangle has legs of length aa and a+1a+1, and hypotenuse a+2a+2 for some positive aa. Find aa and the acute angles (to the nearest tenth of a degree).

Example 34

hard
A right triangle has hypotenuse 2525 and area 8484. Find the legs.

Example 35

hard
A weather balloon is observed from points AA and BB on level ground, 200200 m apart, both on the same side of the balloon's vertical line. Angles of elevation are 40°40° from AA and 60°60° from BB (BB is closer). Find the balloon's height (to the nearest meter).
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Background Knowledge

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

trianglespythagorean theoremratios