SOLVE RIGHT TRIANGLE GUIDE

SOH CAH TOA, explained visually

5 min read · By Justin Rogers · Updated October 5, 2026

SOH CAH TOA is a reminder for the three main trigonometric ratios in a right triangle. Each ratio compares two side lengths relative to a selected acute angle.

SOH: sin(θ) = opposite / hypotenuse
CAH: cos(θ) = adjacent / hypotenuse
TOA: tan(θ) = opposite / adjacent

Label the sides before choosing a formula

Start at the angle you are using. The opposite side is across from it. The adjacent leg touches it. The hypotenuse faces the right angle and is the longest side.

When you select the other acute angle, the opposite and adjacent legs swap. The hypotenuse stays the same. In our diagram, a is opposite A and adjacent to B.

Use the ratio that contains the known and unknown sides

You do not need all three ratios for every problem. If you know the hypotenuse and want the opposite leg, sine connects exactly those measurements. If you know the adjacent leg and want the opposite leg, use tangent.

Example: a 10 m hypotenuse and angle A = 30°

sin(A) = a / c, so a = c × sin(A).

a = 10 × sin(30°) = 10 × 0.5 = 5 m.

cos(A) = b / c, so b = 10 × cos(30°) ≈ 8.66 m.

The remaining angle is B = 60°.

Rearrange before entering numbers

If sin(θ) = opposite / hypotenuse, finding the hypotenuse requires division: hypotenuse = opposite / sin(θ). Multiplication would solve a different problem.

Check whether the answer makes sense

Both acute angles lie strictly between 0° and 90°.
The hypotenuse is longer than either leg.
The sine and cosine of an acute angle are between 0 and 1.
All side lengths use compatible units.

These ratios apply to right triangles. General triangles need different methods.

Explore the 30° example

Formula reference

For an independent explanation of these relationships, see Pearson’s right triangle formulas and worked examples.

Related right triangle guides

Open the right triangle calculator to check your own measurements.