Right Triangle Trigonometry
SOH-CAH-TOA: sine, cosine, and tangent in a right triangle.
The idea
In a right triangle, once you pick a non-right angle θ, its three sides get names relative to θ: the hypotenuse (always opposite the right angle), the opposite side, and the adjacent side. The mnemonic SOH-CAH-TOA connects them:
Use these to find a missing side when you know an angle and one side, or use their inverses (sin⁻¹, cos⁻¹, tan⁻¹) to find a missing angle when you know two sides.
Practice problems
In a right triangle, angle θ = 30° and the hypotenuse is 10. Find the length of the opposite side.
Show answer
opposite = 10 · sin(30°) = 10 × 0.5 = 5.
A right triangle has an opposite side of 7 and an adjacent side of 24. Find θ, rounded to the nearest degree.
Show answer
tan θ = 7/24 ≈ 0.2917, so θ = tan⁻¹(0.2917) ≈ 16°.
A 20-foot ladder leans against a wall, making a 65° angle with the ground. How high does it reach? Round to the nearest tenth of a foot.
Show answer
height = 20 · sin(65°) ≈ 20 × 0.9063 ≈ 18.1 feet.
From a point 50 m from the base of a tower, the angle of elevation to the top is 40°. Find the tower's height, to the nearest tenth of a meter.
Show answer
height = 50 · tan(40°) ≈ 50 × 0.8391 ≈ 42.0 meters.