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

The idea

This row requires you to know and apply Pythagoras’ theorem and the three trigonometric ratios in right-angled triangles in two dimensions. Pythagoras’ theorem states that for a right-angled triangle with hypotenuse c and legs a and b, a² + b² = c². The trigonometric ratios are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, where θ is an angle in the triangle. To find a missing side, substitute two known lengths into the appropriate formula and solve. To find a missing angle, use the inverse trigonometric function on the ratio of the two relevant sides. For example, if the opposite side is 3 cm and the hypotenuse is 5 cm, sin θ = 3/5 = 0.6, so θ = sin⁻¹(0.6) ≈ 36.9°. Always identify the hypotenuse, the side opposite the angle and the side adjacent to the angle before choosing a formula.

What earns the marks

• Correctly state Pythagoras’ theorem as a² + b² = c², where c is the hypotenuse.
• Correctly state sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent.
• Identify the hypotenuse, opposite and adjacent sides correctly for a given angle in a right-angled triangle.
• Substitute known lengths into Pythagoras’ theorem to find a missing side in a two-dimensional right-angled triangle.
• Use a trigonometric ratio and its inverse to find a missing angle in a two-dimensional right-angled triangle.
• Use a trigonometric ratio to find a missing side in a two-dimensional right-angled triangle when one side and one angle are known.

Common mistakes and how to put them right

• Misidentifying the hypotenuse as any slanted side rather than the side opposite the right angle.

  Put it right by locating the right angle first and choosing the side opposite it.

• Using the wrong trigonometric ratio for the given sides.

  Put it right by labelling opposite and adjacent relative to the angle in question before selecting sin, cos or tan.

• Forgetting to use the inverse trigonometric function when finding an angle.

  Put it right by applying sin⁻¹, cos⁻¹ or tan⁻¹ to the ratio.

• Substituting into Pythagoras’ theorem without squaring the sides.

  Put it right by ensuring each term is squared, for example a² + b² = c², not a + b = c.

Exam tips

• Sketch the right-angled triangle and label the hypotenuse, opposite and adjacent sides before choosing a formula.
• Check that your calculator is in degree mode when working with angles in degrees.
• Round final answers to a sensible degree of accuracy, usually three significant figures unless the question states otherwise, and show your substitution clearly.