Pythagoras and trigonometry
Maths · GCSE · The School
Pythagoras: sides only
In a right-angled triangle, a² + b² = c², where c is the hypotenuse — always the longest side, always opposite the right angle. Finding the hypotenuse: add the squares and square-root. Finding a shorter side: SUBTRACT, then square-root. Getting that the wrong way round is the single most common slip, and the check is easy: the hypotenuse must come out longest.
SOH CAH TOA
Label the sides from the angle you are working with: Opposite, Adjacent, and the Hypotenuse. Then sin = O/H, cos = A/H, tan = O/A. Write down which two sides the question gives you and which it wants — that pair names the ratio for you. The labelling, not the algebra, is where marks are lost.
Going backwards for an angle
If you know two sides and want the angle, use the inverse: sin⁻¹, cos⁻¹, tan⁻¹. tan θ = 5/8 gives θ = tan⁻¹(0.625) = 32.0°. Check your calculator is in DEGREES — an answer of 0.558 means it is in radians, and that is a whole-question error worth catching before you write anything down.
A right-angled triangle has hypotenuse 10 cm and one angle of 35°. Find the side OPPOSITE the 35°. You have an angle, so this is trigonometry, not Pythagoras. Opposite and hypotenuse means SIN: sin 35° = O/10, so O = 10 × sin 35° = 5.74 cm. Now find the third side with Pythagoras: 10² − 5.74² = 100 − 32.9 = 67.1, so the side is 8.19 cm. Rule of thumb: an angle anywhere in the question means trigonometry.
Non-right-angled triangles need a different rule
SOH CAH TOA only works inside a right angle. For any other triangle, use the sine rule: a／sin A = b／sin B = c／sin C, matching each side to the angle opposite it. To find a missing side, given angle A = 40°, angle B = 60° and side a = 8 cm: b = a × sin B ÷ sin A = 8 × sin 60° ÷ sin 40° ≈ 10.8 cm. Spotting that a triangle has no right angle is the trigger for reaching for the sine or cosine rule instead of SOH CAH TOA.
Exact values, and the calculator trap
The sine, cosine and tangent of 30, 45 and 60 degrees are expected as exact surds on the non-calculator paper. Also check the calculator is in degrees — a whole question can be lost to radians, and the giveaway is an answer that is wildly small.