Right Triangle Trigonometry

Core Concept

Trig Ratios as Side Ratios

For an acute angle $\theta$ in a right triangle, the three primary trig ratios are defined purely by side lengths:

A compact memory aid: SOH-CAH-TOA.

Why the Ratio Depends Only on the Angle (Similarity)

Any two right triangles sharing an acute angle $\theta$ are similar (AA similarity: the right angle accounts for one pair, $\theta$ for the second). Similar triangles have proportional corresponding sides, so the ratio $\frac{\text{opposite}}{\text{hypotenuse}}$ is the same in both triangles — it doesn't matter how large the triangle is. This is why $\sin\theta$, $\cos\theta$, and $\tan\theta$ are well-defined functions of the angle alone.

Example. A right triangle has legs 3 and 4 with hypotenuse 5. A second right triangle shares the same acute angle $\theta$ (opposite the leg of length 3) and has hypotenuse 15. By similarity, the opposite side in the second triangle is $15 \times \dfrac{3}{5} = 9$. You can verify: $\dfrac{9}{15} = \dfrac{3}{5}$. ✓

Finding a Missing Side

Identify the angle, then label the sides as opposite, adjacent, or hypotenuse relative to that angle. Pick the ratio that involves the two sides you care about, then solve.

Complementary Angle Identity

In any right triangle the two acute angles sum to 90°. If one angle is $x°$, the other is $(90 - x)°$. The side opposite $x°$ is the side adjacent to $(90 - x)°$, and both share the same hypotenuse. Therefore:

This also means $\cos x° = \sin(90° - x°)$.

Numerical example. $\cos 72° = \sin(90° - 72°) = \sin 18°$. Note that the function changes (cosine becomes sine) and the angle changes to its complement; simply keeping cosine and swapping to 18° gives $\cos 18° \approx 0.951$, which does not equal $\cos 72° \approx 0.309$.