Practice Right Triangle Trigonometry in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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 3

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 4

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 5

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 6

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

Example 7

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

Example 8

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

Example 9

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 10

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

Example 11

medium
If sin⁡θ=22\sin\theta = \dfrac{\sqrt{2}}{2} in a right triangle, what is θ\theta?

Example 12

hard
Simplify sin⁡θcos⁡θ+cos⁡θsin⁡θ\dfrac{\sin\theta}{\cos\theta} + \dfrac{\cos\theta}{\sin\theta}.

Example 13

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 14

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

Example 15

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

Example 16

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

Example 17

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 18

easy
Find sin⁡60°\sin 60° exactly.

Example 19

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

Example 20

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