The sine and cosine formulae

Q: Do I need to memorize the cosine rule for the exam?

No, the cosine rule is printed on the official exam formula sheet. You only need to memorize the sine rule and triangle area formula .

Q: When do I use the sine rule vs the cosine rule?

Use the sine rule when you have a matching pair of known side + opposite angle. Use the cosine rule when you have either two sides and their included angle, or all three sides of the triangle.

Edexcel International GCSE Further Pure Mathematics· S10 10D· 25 min read

1. The Sine Rule★★☆☆☆⏱ 7 min

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📘 Definition

Sine Rule

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

Relates the lengths of sides of any triangle to the sines of their opposite angles. This formula must be recalled for the exam, it is not provided on the formula sheet.

The sine rule is used when you are given at least one pair of corresponding side and opposite angle, plus one additional side or angle. Always label your triangle consistently: side $a$ is opposite angle $A$ , side $b$ opposite angle $B$ , side $c$ opposite angle $C$ .

📐 Worked Example

In triangle $A B C$ , angle $A = 4 0^{\circ}$ , side $a = 8$ cm, side $b = 10$ cm. Find angle $B$ , correct to 1 decimal place.

1
Substitute known values into the sine rule formula:
2
$\frac{8}{\sin 40^\circ} = \frac{10}{\sin B}$
3
Rearrange to isolate $sin B$ :
4
$\sin B = \frac{10 \times \sin 40^\circ}{8}$
5
Calculate the right-hand side: $sin B = \frac{10 \times 0.6428}{8} = 0.8035$
6
Take inverse sine to find $B$ : $B = arcsin (0.8035) = 53. 5^{\circ}$ (1 d.p.)
7
Note: This is the ambiguous case, so a second valid solution is $B = 18 0^{\circ} - 53. 5^{\circ} = 126. 5^{\circ}$ , check if this fits the problem context.

Exam tip:

If the question does not specify which triangle to use, give both possible solutions for the ambiguous case to earn full marks.

2. The Cosine Rule★★★☆☆⏱ 7 min

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📘 Definition

Cosine Rule

a^2 = b^2 + c^2 - 2bc \cos A

Used to find unknown sides or angles in triangles where the sine rule cannot be applied. This formula is provided on the exam formula sheet, you do not need to memorize it.

Use the cosine rule in two scenarios: 1) You know two sides and the included angle between them, and need to find the third side; 2) You know all three sides of the triangle, and need to find any interior angle.

📐 Worked Example

In triangle $P QR$ , sides $p = 5$ cm, $q = 7$ cm, included angle $R = 6 0^{\circ}$ . Find the length of side $r$ , correct to 2 significant figures.

1
Label the triangle: side $r$ is opposite angle $R$ , so substitute into the cosine rule:
2
$r^2 = p^2 + q^2 - 2pq \cos R$
3
$r^2 = 5^2 + 7^2 - 2(5)(7)\cos 60^\circ$
4
Calculate each term: $r^{2} = 25 + 49 - 70 \times 0.5 = 74 - 35 = 39$
5
Take square root: $r = 39 = 6.2$ cm (2 s.f.)

Exam tip:

Rearrange the cosine rule to solve for angles directly if needed: $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ to save time in exams.

3. Area of a Triangle: ½ab sin C★★☆☆☆⏱ 6 min

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📘 Definition

Trigonometric Triangle Area Formula

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

Calculates the area of any triangle when you know the lengths of two sides and the size of the included angle between them. This formula must be recalled for the exam, it is not provided on the formula sheet.

This formula works for all triangles, not just right-angled ones. The angle $C$ must always be the included angle between sides $a$ and $b$ for the formula to work correctly.

📐 Worked Example

Find the area of triangle $X Y Z$ where sides $x = 12$ cm, $z = 9$ cm, included angle $Y = 3 0^{\circ}$ . Give your answer as an exact value.

1
Identify the two sides and included angle: sides $x$ and $z$ meet at angle $Y$ , so substitute into the formula:
2
$\text{Area} = \frac{1}{2} \times x \times z \times \sin Y$
3
$\text{Area} = \frac{1}{2} \times 12 \times 9 \times \sin 30^\circ$
4
Use exact value $sin 3 0^{\circ} = 0.5$ :
5
$\text{Area} = 54 \times 0.5 = 27 \text{ cm}^2$

Exam tip:

If the question asks for an exact area, use the exact trigonometric values for 30°, 45°, 60° instead of decimal approximations to avoid losing marks.

4. Combining Formulae for Exam Problems★★★★☆⏱ 7 min

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Most exam questions will require you to use multiple formulae in sequence to solve a full problem. Always start by labeling your triangle clearly, listing all known values, then selecting the correct formula for the first unknown you need to find.

📐 Worked Example

In triangle $A B C$ , $A B = 10$ cm, $B C = 8$ cm, angle $B A C = 5 0^{\circ}$ . Find the area of triangle $A B C$ , correct to 1 decimal place.

1
First, use the sine rule to find angle $A C B$ :
2
$\frac{BC}{\sin BAC} = \frac{AB}{\sin ACB}$
3
$\frac{8}{\sin 50^\circ} = \frac{10}{\sin C} \implies \sin C = \frac{10 \times 0.7660}{8} = 0.9575$
4
Two possible values for $C$ : $73. 2^{\circ}$ or $106. 8^{\circ}$
5
First solution: angle $B = 180 - 50 - 73.2 = 56. 8^{\circ}$ , area = $\frac{1}{2} \times 8 \times 10 \times sin 56. 8^{\circ} = 33.5$ cm²
6
Second solution: angle $B = 180 - 50 - 106.8 = 23. 2^{\circ}$ , area = $\frac{1}{2} \times 8 \times 10 \times sin 23. 2^{\circ} = 15.8$ cm²

Exam tip:

Show all intermediate steps when combining formulae: examiners award marks for correct application of each individual formula even if your final answer is wrong.

5. Common Pitfalls

Wrong move:

Using the sine rule when only sides and no opposite angle pairs are known

Why:

The sine rule requires at least one matching side-angle pair to function, leading to incorrect equations if this condition is not met

Correct move:

Use the cosine rule instead when you have two sides and included angle, or all three sides

Wrong move:

Forgetting to check for the ambiguous case when using the sine rule with SSA measurements

Why:

Two valid triangles can exist, leading to missing half the solution and losing marks

Correct move:

Always calculate the supplementary angle ( $18 0^{\circ} -$ calculated angle) and verify if it is a valid solution (sum of angles < 180°)

Wrong move:

Using a non-included angle in the ½ab sin C area formula

Why:

The formula only works if the angle is between the two sides used, leading to incorrect area values

Correct move:

If you do not have the included angle, use the sine/cosine rule to find it first before calculating area

Wrong move:

Trying to memorize the cosine rule when it is provided on the formula sheet

Why:

Wastes valuable memorization capacity for other required formulae like the sine rule and area formula

Correct move:

Familiarize yourself with the formula sheet layout to quickly locate the cosine rule during exams

Wrong move:

Mixing up side and opposite angle labels when substituting into formulae

Why:

This leads to incorrect values for sides and angles, even if you applied the formula structure correctly

Correct move:

Always label your triangle with side $a$ opposite angle $A$ , $b$ opposite $B$ , $c$ opposite $C$ before substituting any values

6. Quick Reference Cheatsheet

Formula	Use Case	Given on Formula Sheet?
$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$	Find unknown side/angle, 1+ side-angle pair known	No
$a^{2} = b^{2} + c^{2} - 2 b c cos A$	Find 3rd side (2 sides + included angle) / find angle (all 3 sides known)	Yes
$Area = \frac{1}{2} ab sin C$	Calculate area, 2 sides + included angle known	No

7. Frequently Asked

Do I need to memorize the cosine rule for the exam?

No, the cosine rule $a^{2} = b^{2} + c^{2} - 2 b c cos A$ is printed on the official exam formula sheet. You only need to memorize the sine rule $\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$ and triangle area formula $\frac{1}{2} ab sin C$ .

When do I use the sine rule vs the cosine rule?

Use the sine rule when you have a matching pair of known side + opposite angle. Use the cosine rule when you have either two sides and their included angle, or all three sides of the triangle.

Going deeper

formula_sheetEdexcel 4PM1 Further Pure Math Formula SheetCosine rule is provided on this official exam sheet

What's Next

Now that you have mastered the sine and cosine formulae, you can apply these skills to solve 2D and 3D trigonometry problems, the next sub-topic in the Edexcel IGCSE Further Pure Math syllabus. You will also use these formulae alongside coordinate geometry and vector problems later in the course. Practise past exam questions to familiarize yourself with how these formulae are tested in structured written papers, and make sure you can quickly recall the sine rule and area formula under timed conditions. Remember to always check for the ambiguous case in SSA problems, as this is a common mark-losing trap in exams.