G22–G23Geometry & MeasuresHigherGrades 7–9

Sine & Cosine Rules

Q: Do the rules work for right-angled triangles?

Yes. With a 90^ angle the cosine rule becomes Pythagoras, since 90^ = 0.

Q: Can the sine rule give two answers?

When finding an angle it can, because x = (180^ - x). Check whether an obtuse answer is possible.

SOHCAHTOA only works in right-angled triangles. The sine rule, the cosine rule and the area formula $\frac{1}{2} ab sin C$ work in every triangle. They let you find missing sides, angles and areas in surveying, navigation and 3D problems.

50 min AQA GCSE Maths

The big picture

Label the triangle with capital letters for angles and matching lower-case letters for the opposite sides: side $a$ is opposite angle $A$ . With that labelling, every formula is easy to state and check.

The choice of rule depends on what you know. A side and its opposite angle as a pair means the sine rule. Two sides and the angle between them, or all three sides, means the cosine rule.

What you'll learn

Use the sine rule to find a missing side or angle
Use the cosine rule to find a missing side or angle
Find the area of a triangle using $\frac{1}{2} ab sin C$
Choose the correct rule and solve multi-step problems

The sine rule

$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ . For angles, flip it: $\frac{sin A}{a} = \frac{sin B}{b}$ .

Use it when you know a side and its opposite angle.

\frac{b}{sin 7 5 ^{\circ}} = \frac{8}{sin 4 0 ^{\circ}}

b = \frac{8 sin 7 5 ^{\circ}}{sin 4 0 ^{\circ}} = 12.02 \dots

Answer

b = 12.0

cm (3 s.f.)

Your turnSame method, new numbers — try it before moving on.

In triangle $A B C$ , $A = 3 5^{\circ}$ , $B = 7 0^{\circ}$ and $a = 6$ cm. Find $b$ to 1 decimal place.

Your answer

The cosine rule

For a side: $a^{2} = b^{2} + c^{2} - 2 b c cos A$ .

For an angle: $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ .

Use it with two sides and the included angle, or with three sides.

x^{2} = 9^{2} + 1 4^{2} - 2 (9) (14) cos 6 5^{\circ}

x^{2} = 277 - 106.5 \dots = 170.5 \dots

Answer

x = 13.1

cm (3 s.f.)

Your turnSame method, new numbers — try it before moving on.

Two sides of a triangle are 7 cm and 11 cm, with an angle of $4 8^{\circ}$ between them. Find the third side to 1 decimal place.

Your answer

Why this works

A negative cosine means the angle is obtuse. The cosine rule handles this automatically, which is why it is the safer choice for the largest angle of a triangle.

Area of a triangle

Area $= \frac{1}{2} ab sin C$ , where $C$ is the angle between sides $a$ and $b$ .

1Area

= \frac{1}{2} \times 7 \times 10 \times sin 11 5^{\circ}

Answer

31.7

m² (3 s.f.)

Your turnSame method, new numbers — try it before moving on.

Find the area of a triangle with sides 8 cm and 12 cm and an included angle of $3 0^{\circ}$ , in cm².

Your answer

Tip — Keep calculator values unrounded between steps and round only the final answer.

Think like an examiner

Write the formula with values substituted before calculating.
Check your calculator is in degree mode.
In multi-step problems, state which triangle and rule you are using at each step.

Remember these

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

Sine rule.

a^{2} = b^{2} + c^{2} - 2 b c cos A

Cosine rule.

Area = \frac{1}{2} ab sin C

Area of any triangle.

Watch out for these

Calculating

b^{2} + c^{2} - 2 b c

first and then multiplying by

cos A

Work out

2 b c cos A

as one term, then subtract.

Using the sine rule without a side–angle pair.

With two sides and the included angle, use the cosine rule.

Using a side that is not next to the angle in

\frac{1}{2} ab sin C

The angle must be between the two sides.

Rounding an intermediate side, then using it again.

Use the stored calculator value.

Stretch yourself

A parallelogram has sides 6 cm and 9 cm with an angle of $7 0^{\circ}$ between them. Find its area and the lengths of both diagonals.

Hint — The parallelogram is two congruent triangles. Adjacent angles add to $18 0^{\circ}$ .

Questions students ask

Do the rules work for right-angled triangles?

Yes. With a

9 0^{\circ}

angle the cosine rule becomes Pythagoras, since

cos 9 0^{\circ} = 0

Which form of the sine rule should I use?

Put the unknown on top: sides on top to find a side, sines on top to find an angle.

Can the sine rule give two answers?

When finding an angle it can, because

sin x = sin (18 0^{\circ} - x)

. Check whether an obtuse answer is possible.

Key takeaways

Side $a$ is opposite angle $A$ .
Sine rule: when you have a side and its opposite angle.
Cosine rule: two sides and the included angle, or three sides.
Area $= \frac{1}{2} ab sin C$ with the included angle.

Exam-style questions

Practise it the way it is examined

3 original questions · 11 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 75 marks

In triangle $P QR$ , $P Q = 9$ cm, $QR = 12$ cm and angle $P QR = 10 5^{\circ}$ .

(a)Work out the length of $P R$ . Give your answer to 3 significant figures.[3]
(b)Work out the area of the triangle.[2]

Question 2Grade 83 marks

A triangle has sides of 6 cm, 9 cm and 13 cm. Work out the size of its largest angle.[3]

Question 3Grade 93 marks

A triangle has sides of 8 cm and 10 cm and an area of 30 cm². The angle between these sides is acute. Find the angle.[3]

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Sine & Cosine Rules

The sine rule

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