Trigonometry and Pythagoras' theorem

• Pythagoras: a² + b² = c², where c is the hypotenuse.
• sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj, for right-angled triangles.
• Use these to solve 2D problems, including bearings.
• Higher: sine and cosine of obtuse angles; angles of elevation and depression.
• Higher: sine rule, cosine rule and area = ½ab sin C for any triangle (given on the formulae sheet).
• Higher: Pythagoras and trigonometry in 3D, including the angle between a line and a plane.

The three sides of a right-angled triangle

The hypotenuse is the longest side, opposite the right angle. To find it, add the squares; to find a shorter side, subtract. Pythagoras is not on the formulae sheet.

SOHCAHTOA

Label the sides relative to the angle θ: opposite (across from θ), adjacent (next to θ, not the hypotenuse), hypotenuse. Choose the ratio that uses the two sides you have or want. To find an angle, use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹). These ratios must be memorised. Make sure the calculator is in degrees.

Draw it, find the right angle

Draw a sketch with North lines. Bearings questions usually hide a right-angled triangle formed by the North–South and East–West directions.

Beyond 90°; looking up and down

For obtuse angles: sin(180° − x) = sin x and cos(180° − x) = −cos x. So cos of an obtuse angle is negative, and sin⁻¹ on a calculator gives only the acute answer — the obtuse answer is 180° minus it.

An angle of elevation is measured up from the horizontal; an angle of depression is measured down from the horizontal. The angle of depression from A to B equals the angle of elevation from B to A (alternate angles).