Right-triangle trig · SAT Maths

Concept

For an acute angle θ in a right triangle:

SOH — sin θ = opposite / hypotenuse
CAH — cos θ = adjacent / hypotenuse
TOA — tan θ = opposite / adjacent

Complementary-angle identity: sin θ = cos(90° − θ). The two acute angles in a right triangle add to 90°, and they swap their opposite/adjacent roles.

Use inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) to recover an angle when you know a ratio.

Worked example 1

A right triangle has angle θ with opposite side 3 and hypotenuse 5. Find sin θ, cos θ, and tan θ.

Solution

sin. sin θ = 3/5

Adjacent. By Pythagoras: adj = √(5² − 3²) = √16 = 4

cos & tan. cos θ = 4/5, tan θ = 3/4

sin θ = 3/5, cos θ = 4/5, tan θ = 3/4. (Classic 3-4-5 right triangle.)

Worked example 2

A ramp rises 4 feet over a horizontal run of 9 feet. What angle (to the nearest degree) does it make with the ground?

Solution

Setup. Opposite (rise) = 4, adjacent (run) = 9.

tan. tan θ = 4/9

Inverse. θ = tan⁻¹(4/9) ≈ 24°

Approximately 24°.

Practice test

8 questions on SOH-CAH-TOA, complementary identities, and inverse trig.

Practice test · 8 questions Question 1 of 8 · Score 0