Key Stage 4 maths · Geometry and measures

GCSE Pythagoras, trigonometry and exact values: a parent’s guide

Right-angled triangles are at the heart of GCSE geometry. Children use Pythagoras’ Theorem to find missing sides, and the sine, cosine and tangent ratios to find missing sides and angles in right-angled triangles in two dimensions. They also learn the exact values of sin and cos for 0°, 30°, 45°, 60° and 90°, and of tan for 0°, 30°, 45° and 60°, which are needed when a calculator is not allowed.

Pythagoras and trigonometry, explained

Pythagoras’ Theorem says that in a right-angled triangle the square of the longest side, the hypotenuse, equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is always opposite the right angle. To find the hypotenuse, children add the squares and take the square root. To find a shorter side they subtract. Pythagoras is also used for the distance between two points on a coordinate grid and for checking whether a triangle contains a right angle at all.

Trigonometry links the angles of a right-angled triangle to the ratios of its sides. Children first label the sides relative to the angle they are interested in: the hypotenuse, the opposite side across from the angle, and the adjacent side next to it. Then sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent, remembered as SOHCAHTOA. They choose the ratio that involves the two sides they have or need. To find a side they rearrange the equation. To find an angle they use the inverse function on the calculator, written sin⁻¹, cos⁻¹ or tan⁻¹, with the calculator set to degrees.

Some values of sin, cos and tan can be given exactly, with no decimals. sin 30° = 1/2, cos 60° = 1/2, sin 60° = cos 30° = √3/2, sin 45° = cos 45° = √2/2 (also written 1/√2), sin 0° = 0, cos 0° = 1, sin 90° = 1 and cos 90° = 0. For tangent, tan 0° = 0, tan 30° = 1/√3, tan 45° = 1 and tan 60° = √3. Tan 90° is not defined, so it isn’t asked for. It helps to see where the table comes from instead of memorising it blind: halving an equilateral triangle of side 2 gives a triangle with sides 1, √3 and 2, and halving a square gives one with sides 1, 1 and √2. All the exact values can be read off those two triangles.

In real problems the triangle usually has to be found first. Typical contexts include ladders against walls, ramps, angles of elevation and depression, and bearings. An isosceles triangle is often split down the middle into two right-angled triangles. The best habit is to draw a clear sketch, mark what is known, and keep every digit on the calculator until the final line. If a question doesn’t say how to round, lengths are often given to three significant figures and angles to one decimal place.

Words your child will hear

Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Opposite: The side directly across from the angle you are working with.
Adjacent: The side next to the angle you are working with that is not the hypotenuse.
Sine, cosine, tangent: Three ratios of sides in a right-angled triangle, shortened to sin, cos and tan.
Inverse function: The button that turns a ratio back into an angle, such as cos⁻¹(0.3) ≈ 72.5°.
Surd: A root that cannot be written as a whole number or fraction, such as √3, left exact in an answer.
Angle of elevation: The angle upwards from the horizontal to something you are looking up at.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles in 2 dimensional figures
Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°
Know the exact value of tan θ for θ = 0°, 30°, 45°, 60°

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A 5 m ladder leans against a vertical wall with its foot 1.5 m from the wall. How far up the wall does it reach, and what angle does the ladder make with the ground?

Sketch a right-angled triangle. The ladder is the hypotenuse (5 m), the ground is the base (1.5 m) and the wall is the height, h.
Find the height with Pythagoras. The height is a shorter side, so subtract.
```
h² + 1.5² = 5²
h² = 25 − 2.25 = 22.75
h = √22.75 = 4.77 m (3 s.f.)
```
For the angle θ at the ground, label the sides from that angle. The 1.5 m is next to θ so it is adjacent, and the 5 m ladder is the hypotenuse. Adjacent and hypotenuse means cosine.

Write the ratio, then use the inverse function with the calculator in degrees.

cos θ = adjacent ÷ hypotenuse
cos θ = 1.5 ÷ 5 = 0.3
θ = cos⁻¹(0.3) = 72.5° (1 d.p.)

Check with the other ratio. sin 72.5° is about 0.954 and 4.77 ÷ 5 is about 0.954, so both parts agree.

The ladder reaches 4.77 m up the wall and makes an angle of 72.5° with the ground.

Common mistakes and how to help

Adding the squares when looking for a shorter side

The first Pythagoras questions all ask for the hypotenuse, so “add the squares and square root” becomes automatic. With a hypotenuse of 13 and a side of 5, adding gives √194 ≈ 13.9, which is longer than the hypotenuse.

What helps: Ask whether the answer should be longer or shorter than the sides you were given. The hypotenuse is always the longest side, so a shorter side must come from subtracting.

Labelling opposite and adjacent from the wrong angle

Children remember “opposite is the one at the top” or “adjacent is the bottom”. The labels depend on which angle is being used, so they change if the triangle is turned or a different angle is chosen.

What helps: Always label in the same order: hypotenuse first, then opposite (the side that doesn’t touch the angle), then adjacent as the one left over. Redo the labels whenever the angle changes.

Calculator slips: wrong mode, or the wrong button for an angle

A calculator left in radians gives wrong answers that look plausible. Pressing sin instead of sin⁻¹ when finding an angle gives a ratio, not an angle.

What helps: Test the calculator before each exam question: sin 30 should display 0.5, and sin⁻¹(0.5) should give 30. An angle in a right-angled triangle must be below 90°, so anything else signals a slip.

Mixing up the exact values, such as sin 30° and cos 30°

The table of values looks like a list of arbitrary fractions and roots, so memory swaps them, especially √3/2 and 1/2.

What helps: Sketch the 30°, 60°, 90° triangle with sides 1, √3 and 2 and calculate each ratio from it. Sine rises from 0 to 1 as the angle grows from 0° to 90°, while cosine falls from 1 to 0.

How to help at home

Check a right angle with 3, 4, 5. Mark 30 cm and 40 cm along two edges meeting at a corner, such as a book, box or door frame. If the straight line between the marks is exactly 50 cm, the corner is a right angle. Try 5, 12, 13 as well.
Make a simple clinometer. Tape a straw along the straight edge of a protractor and hang a weighted string from its centre. From a measured distance away in a garden or park, away from roads, look through the straw at the top of a tree or building and read where the string crosses the scale. The gap between the string and the 90° mark is the angle of elevation. Then use tan, remembering to add your eye height. Never look towards the sun.
Fold the special triangles. Draw an equilateral triangle with sides of 2 units, then split it down the middle. Use Pythagoras to find the height. Your child can then read off sin 60° and cos 30° for themselves, which sticks far better than a memorised table. Cut a square of side 1 along its diagonal for the 45° values.
Teach it back. Draw a right-angled triangle, turn it upside down, and ask your child to label the sides from one angle and then from the other. Teaching you the labelling out loud shows quickly where the confusion is.

Builds on

Key Stage 3: Perimeter, area and volume
Key Stage 3: Constructions and scale drawings
Key Stage 3: Transformations and enlargement
Key Stage 3: Angles, parallel lines and polygons
Key Stage 3: Pythagoras and trigonometry