Key Stage 4 maths · Geometry and measures

GCSE sine rule, cosine rule and 3D trigonometry (higher)

Right-angled trigonometry only works when there is a right angle. At the higher tier of GCSE maths, children learn the sine rule, the cosine rule and the formula Area = ½ab sin C, which find lengths, angles and areas in any triangle. They also apply Pythagoras’ Theorem and trigonometry to three-dimensional shapes such as cuboids and pyramids, usually by finding the right-angled triangle hidden inside the solid.

Sine rule and cosine rule, explained

In an ordinary triangle labelled ABC, each side is named with the lower-case letter of the angle opposite it: side a is opposite angle A, and so on. The sine rule says a/sin A = b/sin B = c/sin C. It is used when a side and its opposite angle are known, plus one more piece of information, such as another angle or side. Children use it in the upside-down form sin A/a = sin B/b when they are finding an angle. One thing to watch is that the inverse sine on a calculator never gives an angle above 90°, so when the triangle could contain an obtuse angle, the other possible answer is 180° minus the calculator value.

The cosine rule, a² = b² + c² − 2bc cos A, covers the cases the sine rule cannot: two sides and the angle between them (to find the third side), or all three sides (to find an angle, after rearranging to cos A = (b² + c² − a²) / 2bc). Notice that if A = 90°, cos A = 0 and the rule becomes Pythagoras’ Theorem, so the cosine rule is a more general version of a familiar result. Children learn to choose between the two rules by looking at what they are given, and many find it helps to write down a short list of the known and unknown parts first.

The area of any triangle is ½ab sin C, where a and b are two sides and C is the angle between them. This is useful when the height is not given. It can also be rearranged to find an unknown side or angle when the area is known. The rule only works if the angle is the one included between the two sides used.

In three dimensions, the trick is to pick out a flat triangle, often right-angled, inside the solid and draw it separately. For a cuboid, the diagonal across the base is found first by Pythagoras, then used in a second right-angled triangle with the height to find either the long diagonal or the angle it makes with the base. The same approach finds the angle between a sloping edge of a pyramid and its base. These questions need careful sketches, and children are expected to explain which triangle they are using.

Words your child will hear

Included angle: The angle between two given sides. The area formula needs this one.
Sine rule: a/sin A = b/sin B = c/sin C, linking each side to the angle opposite it.
Cosine rule: a² = b² + c² − 2bc cos A, used with two sides and the included angle, or with three sides.
Opposite (in a general triangle): Side a is opposite angle A, that is, the side that doesn’t touch the angle.
Obtuse: An angle between 90° and 180°.
Space diagonal: A line joining opposite corners of a cuboid through the inside of the shape.
Angle between a line and a plane: The angle between the line and its shadow on the flat surface, if a light shone at right angles onto that surface. It is found in a right-angled triangle.

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 general triangles, where possible
Apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in 3 dimensional figures
Know and apply the sine rule, a/sin A = b/sin B = c/sin C, and cosine rule, a² = b² + c² − 2bc cos A, to find unknown lengths and angles
Know and apply Area = 1/2 ab sin C to calculate the area, sides or angles of any triangle

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

A worked example

In triangle ABC, AB = 8 cm, AC = 11 cm and angle A = 40°. Find the length BC, and the area of the triangle.

Sketch the triangle and label the sides. BC is opposite angle A, so call it a. The sides next to A are b = 11 cm and c = 8 cm. Two sides and the included angle means the cosine rule.

Substitute into a² = b² + c² − 2bc cos A. Work out the 2bc cos A term as a whole, before subtracting.

a² = 11² + 8² − 2 × 11 × 8 × cos 40°
a² = 121 + 64 − 176 × 0.7660...
a² = 185 − 134.82...
a² = 50.17...

Take the square root and round.
```
a = √50.17... = 7.08 cm (3 s.f.)
```
For the area, use ½ab sin C with the two sides and the angle between them, here 8 and 11 and 40°.
```
Area = ½ × 8 × 11 × sin 40°
     = 44 × 0.6427...
     = 28.28... cm²
```
Sense check: the third side must be less than 8 + 11 = 19 cm and more than 11 − 8 = 3 cm, and a narrow angle of 40° gives a fairly short third side, so 7.08 cm is believable.

BC = 7.08 cm (3 s.f.) and the area is 28.3 cm² (3 s.f.).

Common mistakes and how to help

Using the sine rule when the cosine rule is needed, or the other way round

Both look like formulae with sin or cos in them, and many pupils pick the one they remember best. The sine rule needs a matching pair of a side and its opposite angle, which two sides and the included angle don’t provide.

What helps: Before choosing, list what is known. A side and its opposite angle, plus one more value, points to the sine rule. Two sides with the angle between them, or all three sides, points to the cosine rule.

Working out the cosine rule in the wrong order

Typing b² + c² − 2bc into the calculator and then multiplying the whole result by cos A gives a very different number. The subtraction is of the whole term 2bc cos A.

What helps: Calculate 2bc cos A on its own line first, then subtract it from b² + c². Check by seeing whether the answer is plausible: it should be less than the sum of the other two sides.

Using an angle that isn’t between the two sides in Area = ½ab sin C

The formula looks like it will work with any angle, and the triangle may be labelled so that the given angle sits somewhere else.

What helps: Mark the angle and trace the two sides that make it. Those two sides and that angle are the ones that go into the formula.

Failing to draw a right-angled triangle inside a 3D shape

On a 3D diagram, an angle can look right-angled when it is not, and trying to apply the trig ratios straight onto the picture gives wrong results.

What helps: Draw the right-angled triangle separately and label its true lengths. Find any missing base diagonals with Pythagoras first, then use the second triangle for the angle or length wanted.

How to help at home

Check the cosine rule with a drawing. Draw two lines of 8 cm and 11 cm meeting at 40° with a ruler and protractor. Measure the third side and compare it with the answer from the rule (about 7.1 cm). Try another angle, such as 90°, where the rule should give the Pythagoras answer.
Find a space diagonal. Measure a cereal box or shoebox. Use Pythagoras twice to predict the longest straight line that fits inside from corner to opposite corner, then open the box and check by stretching a tape measure from a bottom corner to the opposite top corner. The shortcut is the square root of length² + width² + height², and it is worth checking that it agrees.
Survey a triangle outdoors. In a garden or park, away from roads, mark three points with pegs or sticks and measure all three distances with string and a tape. Use the cosine rule to calculate one angle, then check it by drawing the triangle to scale and measuring that angle with a protractor.
Sort the questions. Write several triangle problems on cards without solving them. Ask your child to sort the cards into sine rule, cosine rule, area formula or right-angled trig. Choosing the method is often the hardest step.

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