Use Pythagoras' theorem and the trigonometric ratios in right-angled triangles, and the sine rule, cosine rule and area formula in any triangle (Higher tier), to find missing lengths and angles.

What this dot point is asking

WJEC asks you to use Pythagoras' theorem and the trigonometric ratios in right-angled triangles to find missing sides and angles, and at Higher tier to use the sine rule, the cosine rule and the area formula in any triangle. The key judgement is choosing the right tool: Pythagoras when only sides are involved, the trig ratios when a right angle and one angle are involved, and the sine or cosine rule when the triangle has no right angle. It is one of the most heavily assessed geometry strands and appears on both components.

Pythagoras' theorem

Pythagoras connects the three sides of a right-angled triangle.

So a triangle with shorter sides $5$ and $12$ has hypotenuse $\sqrt{5^2 + 12^2} = \sqrt{169} = 13$. The same idea finds the distance between two points on a grid by treating the horizontal and vertical gaps as the two shorter sides.

The trigonometric ratios

The ratios link an angle to two of the sides in a right-angled triangle.

Label the sides relative to the angle you are using: the hypotenuse (H) is opposite the right angle, the opposite (O) faces your angle, and the adjacent (A) is next to it. Then SOH CAH TOA gives $\sin\theta = \tfrac{O}{H}$, $\cos\theta = \tfrac{A}{H}$ and $\tan\theta = \tfrac{O}{A}$. To find a side, choose the ratio with the two sides you have, substitute and solve. To find an angle, use the inverse ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) on the ratio of the two known sides.

Choosing trig or Pythagoras

The given information tells you which tool to reach for.

If the question gives only sides and a right angle, use Pythagoras; if it gives an angle and asks for a side (or two sides and asks for an angle) in a right-angled triangle, use the ratios.

The sine and cosine rules (Higher)

When a triangle has no right angle, two rules from the formulae list take over.

A quick decision: if a side is paired with its opposite angle, the sine rule is usually fastest; if not, reach for the cosine rule. Both are printed in the WJEC formulae list, but you must know when each applies.

Why this matters

Pythagoras and trigonometry are among the most frequently examined geometry topics and combine with mensuration, bearings and vectors on harder questions. The marks reward selecting the correct method, labelling sides accurately and rounding sensibly, often to a stated number of decimal places or significant figures. At Higher tier the sine and cosine rules extend the same reasoning to any triangle, and recognising which rule fits the given information is the marked judgement that separates full marks from a stalled attempt.