Pythagoras and trigonometry: finding sides and angles

Right-angled triangles come with two tools, and the first skill is knowing which to reach for. The rule is simple: is there an angle involved (other than the right angle)? No angle → Pythagoras. Yes angle → trigonometry.

Pythagoras: sides only

When you know two sides and want the third, and no angle is mentioned:

where $c$ is the hypotenuse — the longest side, always opposite the right angle. To find a shorter side instead, rearrange: $a^2 = c^2 - b^2$.

Find the hypotenuse of a triangle with legs $3$ and $4$:

Trigonometry: when an angle is involved

If a question gives or asks for an angle, you need the three ratios. Label the sides relative to the angle $\theta$: opposite (facing it), adjacent (beside it), and the hypotenuse. Then SOH-CAH-TOA:

Choosing the ratio

Label the two sides involved in the question, then pick the ratio that uses exactly those two:

1. Mark the angle. Label the sides O, A, H.
2. See which two sides the question gives or wants.
3. Choose sin, cos, or tan accordingly (the pair of letters tells you).

Find the opposite side when $\theta = 30°$ and the hypotenuse is $10$. You have H, you want O → that's $\sin$:

Finding an angle

If you know two sides and want the angle, use the inverse functions ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$). With opposite $= 4$ and adjacent $= 3$:

Find the right angle first, then the hypotenuse. The hypotenuse is always opposite the right angle — never one of the sides forming it. Mislabel it and every ratio after goes wrong.

Last revised 28 June 2025.