Geometry and measures - Mensuration and calculation: know the formulae for: Pythagoras’ theorem a² + b² = c² , and the trigonometric ratios, sin θ = opposite/hypotenuse , cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent ; apply them to find angles and lengths in right-angled triangles in two-dimensional figures (Foundation tier)

The idea

In a right-angled triangle the hypotenuse is the longest side, opposite the right angle, and a² + b² = c² ties the three sides together with c as the hypotenuse. To find the hypotenuse, square the two shorter sides, add, then take the square root: 5 cm and 12 cm give 25 + 144 = 169, so the hypotenuse is 13 cm. To find a shorter side, subtract instead of adding. The trigonometric ratios link an angle to two sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. Label the sides from the angle you are working with, pick the ratio holding the side you know and the side you want, then rearrange. With hypotenuse 10 cm and angle 30°, the opposite side is 10 × sin 30° = 5 cm. To find an angle from two known sides, use the inverse trigonometric function: if opposite = 3 cm and hypotenuse = 6 cm, then sin θ = 3/6 = 0.5, so θ = sin⁻¹(0.5) = 30°.

What earns the marks

• The theorem written with the numbers in place, such as 5² + 12² as the start of a hypotenuse calculation.
• Taking the square root at the end, or subtracting rather than adding when the missing side is a shorter one.
• Choosing the trigonometric ratio that matches the sides labelled in the question.
• A correct rearrangement, and a further mark for the length or angle rounded as the question asks.
• Using the inverse trigonometric function (sin⁻¹, cos⁻¹ or tan⁻¹) to find an angle from two known sides.

Common mistakes and how to put them right

• Adding the squares when the missing side is a shorter one, which always gives a length that is too large.

• Stopping at 169 and never taking the square root, so the side is given as the square of itself.

• Labelling opposite and adjacent from the right angle rather than from the angle named in the question.

• Leaving the calculator in radian mode, so an inverse trigonometric function returns a number far smaller than the angle in degrees.

• Using the ratio itself as the angle instead of applying the inverse trigonometric function, for example writing θ = 0.5 rather than θ = sin⁻¹(0.5) = 30°.

Exam tips

• Mark the hypotenuse on the diagram first; it is always opposite the right angle and never one of the arms of it.
• Write the ratio out in full before you put numbers in, then rearrange once the substitution is on the page.
• Check the answer is possible: the hypotenuse must be the longest side, and each other angle must be under 90°.
• When finding an angle, check that the inverse trigonometric function gives a sensible acute angle for the triangle.