Use Pythagoras' theorem and the trigonometric ratios in right-angled triangles; and apply the sine rule, cosine rule and the area formula in any triangle (Higher tier).

What this dot point is asking

The Eduqas geometry content asks you to use Pythagoras' theorem and the three trigonometric ratios in right-angled triangles at both tiers, and at Higher tier to apply the sine rule, cosine rule and the triangle area formula in any triangle. Pythagoras finds a missing side from the other two; trigonometry links sides and angles. These are among the most useful tools on the paper, feeding into bearings, area, vectors and 3D problems. The right-angled work is core; the sine and cosine rules are reliable Higher-tier multi-mark questions, and they are printed on the Eduqas formulae list.

Pythagoras' theorem

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

So to find a shorter side when the hypotenuse is $13$ and one side is $5$, compute $\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$. The direction of the calculation matters: add the squares for the hypotenuse, subtract for a shorter side. On the non-calculator Component 1, answers are often left in surd form, such as $\sqrt{20} = 2\sqrt{5}$.

The trigonometric ratios

Trigonometry connects an angle to the ratio of two sides, using SOH CAH TOA.

To find a missing side, choose the ratio that uses the side you know and the side you want, then rearrange. To find a missing angle, use the inverse function: if $\tan\theta = \tfrac{3}{4}$, then $\theta = \tan^{-1}\left(\tfrac{3}{4}\right) \approx 36.9^\circ$. Choosing the correct ratio from the labelled sides is the key decision.

The sine and cosine rules (Higher)

For triangles without a right angle, two rules extend trigonometry, both on the formulae list.

The choice depends on what you know: use the sine rule when you have a matching side-angle pair, and the cosine rule when you have two sides and the included angle (to find the third side) or all three sides (to find an angle).

Why this matters

Pythagoras and trigonometry turn geometric diagrams into solvable equations, which is why they recur in bearings, area, vectors and 3D problems across both components. Eduqas often expects exact surd answers on the non-calculator paper and accurate calculator work on the other, and the special-angle values ($\sin 30^\circ = \tfrac{1}{2}$, $\tan 45^\circ = 1$, $\cos 60^\circ = \tfrac{1}{2}$) are worth knowing because they appear without a calculator.