Pythagoras and right-angled trigonometry

Pythagoras' theorem

For a right-angled triangle, a² + b² = c², where c is the hypotenuse — the side opposite the right angle, and always the longest.

• Finding the hypotenuse: add the squares, then square root.
• Finding a shorter side: subtract the squares, then square root.

Getting this the wrong way round produces an impossible triangle, so sense-check: the hypotenuse must be the largest value.

The trigonometric ratios — SOHCAHTOA

Label the sides relative to the angle you are using:

• Hypotenuse — opposite the right angle (fixed).
• Opposite — opposite your angle.
• Adjacent — next to your angle.

Then:

• sin θ = O/H
• cos θ = A/H
• tan θ = O/A

Choosing which ratio

Mark the side you know and the side you want, then pick the ratio containing exactly those two. This is the whole decision, and doing it explicitly prevents most errors.

Finding an angle

Use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹. If sin θ = 0.5 then θ = sin⁻¹(0.5) = 30°.

Angles of elevation and depression

The angle of elevation is measured upward from the horizontal, the angle of depression downward from the horizontal. Both are measured from the horizontal, never from the vertical.