Sine rule, cosine rule and area of triangles

IB Mathematics: Analysis and Approaches SL· Unit 3: Geometry & Trigonometry· 15 min read

1. The Sine Rule★★☆☆☆⏱ 4 min

📘 Definition

Sine Rule

For triangle $A B C$ with sides $a, b, c$ opposite angles $A, B, C$

The ratio of each side to the sine of its opposite angle is constant for all sides and angles in the triangle

Example:

If $a = 5$ , $A = 3 0^{\circ}$ , $B = 6 0^{\circ}$ , we can solve for $b$

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

The sine rule is used when you have either (1) two angles and one side, or (2) two sides and a non-included angle.

📐 Worked Example

In triangle $A B C$ , $A = 4 0^{\circ}$ , $B = 6 0^{\circ}$ , $a = 8$ cm. Find the length of side $b$ .

1
Write the sine rule for the known values:
$\frac{a}{\sin A} = \frac{b}{\sin B}$
2
Substitute the known values:
$\frac{8}{\sin 40^\circ} = \frac{b}{\sin 60^\circ}$
3
Rearrange to isolate $b$ :
$b = \frac{8 \times \sin 60^\circ}{\sin 40^\circ}$
4
Calculate and round to 3 significant figures:
$b \approx 10.8 \text{ cm}$

Exam tip:

Always label sides opposite their corresponding angles to avoid mixing up values.

2. Ambiguous Case of the Sine Rule★★★★☆⏱ 5 min

The ambiguous case only occurs when you have side-side-angle (SSA) information (two sides, non-included angle) where the given angle is acute. There can be 0, 1, or 2 valid triangles that fit the information.

📘 Definition

Ambiguous Case

The scenario where two distinct valid triangles can be formed from the same SSA input, resulting in two correct solutions

📐 Worked Example

Given triangle $A B C$ with $a = 7$ , $b = 9$ , $A = 3 5^{\circ}$ , find all possible measures of angle $B$ .

1
Apply the sine rule:
$\frac{\sin B}{b} = \frac{\sin A}{a}$
2
Substitute values to solve for $oldsymbol{\sin B}$ :
$\sin B = \frac{9 \times \sin 35^\circ}{7} \approx 0.738$
3
Use the identity $oldsymbol{\sin \theta = \sin (180^\circ - \theta)}$ to get both solutions:
$B_1 = \arcsin(0.738) \approx 47.6^\circ, \quad B_2 = 180^\circ - 47.6^\circ = 132.4^\circ$
4
Check validity: $3 5^{\circ} + 132. 4^{\circ} = 167. 4^{\circ} < 18 0^{\circ}$ , so both are valid.

3. The Cosine Rule★★★☆☆⏱ 4 min

📘 Definition

Cosine Rule

For triangle $A B C$ with sides $a, b, c$ opposite angles $A, B, C$

Relates the three sides of a triangle to one of its angles, with two common forms for finding sides or angles

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

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

Use the cosine rule when you have (1) three sides and need to find an angle, or (2) two sides and their included angle and need to find the third side. There is no ambiguous case with cosine rule.

📐 Worked Example

In triangle $A B C$ , $b = 4$ cm, $c = 6$ cm, included angle $A = 5 0^{\circ}$ . Find the length of side $a$ .

1
Use the side form of cosine rule for two sides and included angle:
$a^2 = b^2 + c^2 - 2bc \cos A$
2
Substitute the known values:
$a^2 = 4^2 + 6^2 - 2(4)(6)\cos 50^\circ$
3
Calculate the right-hand side:
$a^2 = 16 + 36 - 48(0.6428) \approx 21.15$
4
Take the square root and round to 3 significant figures:
$a \approx 4.60 \text{ cm}$

Exam tip:

When finding an angle with cosine rule, the sign of $oldsymbol{\cos A}$ tells you if the angle is acute ( $cos A > 0$ ) or obtuse ( $cos A < 0$ ), no extra checks needed.

4. Area of Any Triangle★★☆☆☆⏱ 2 min

📘 Definition

Area of an Oblique Triangle

The area of any triangle (right or oblique) equals half the product of two sides multiplied by the sine of their included angle

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

If you do not know the included angle, you can use the sine or cosine rule to find it first, then apply this formula.

📐 Worked Example

Find the area of triangle $X Y Z$ where $X Y = 5$ cm, $X Z = 7$ cm, angle at $X$ is $3 0^{\circ}$ .

1
Identify the two sides and their included angle: $X Y = 5$ , $X Z = 7$ , included angle $X = 3 0^{\circ}$
2
Substitute into the area formula:
$Area = \frac{1}{2} \times 5 \times 7 \times \sin 30^\circ$
3
Calculate, using $oldsymbol{\sin 30^\circ = 0.5}$ :
$Area = 8.75 \text{ cm}^2$

5. Common Pitfalls

Wrong move:

Mixing up sides and angles in sine rule, using $a / sin B$ instead of $a / sin A$

Why:

Sine rule requires each side divided by the sine of the angle opposite it, not adjacent.

Correct move:

Label sides with lowercase letters matching the uppercase angle opposite them before starting calculations.

Wrong move:

Forgetting to check for a second solution in the ambiguous SSA case

Why:

Most students only calculate the acute solution, missing the valid obtuse solution and losing marks.

Correct move:

Always calculate $18 0^{\circ} - arcsin (x)$ and check if total angles sum to less than $18 0^{\circ}$ to see if it is valid.

Wrong move:

Using sine rule when given two sides and their included angle

Why:

You do not have a side-opposite-angle pair to apply the sine rule directly here.

Correct move:

Use cosine rule first to find the third side, then use sine rule if you need additional angles.

Wrong move:

Applying $\frac{1}{2} ab sin C$ with a non-included angle

Why:

The formula only works when the angle is between the two sides you are using.

Correct move:

Find the included angle first using sine or cosine rule before calculating area.

Wrong move:

Rounding intermediate steps too early, leading to inaccurate final answers

Why:

Early rounding introduces cumulative error that can change the final answer enough to lose accuracy marks.

Correct move:

Keep full precision on your calculator until the final step, then round to the required number of significant figures.

6. Quick Reference Cheatsheet

Scenario	Rule/Formula to Use
Two angles + one side	Sine rule
Two sides + non-included angle (SSA)	Sine rule (check for 2 solutions)
Two sides + included angle (SAS)	Cosine rule for missing side, then $\frac{1}{2} ab sin C$ for area
Three sides (SSS)	Cosine rule for unknown angles
Area of any triangle (SAS given)	$\frac{1}{2} ab sin C$

7. Frequently Asked

When do I use sine rule vs cosine rule?

Use sine rule when you have a matching side and opposite angle pair plus one other side/angle. Use cosine rule when you have three sides, or two sides and their included angle, with no side-opposite-angle pair.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Find missing side and triangle area
2024 · 2
Solve ambiguous sine rule problem
2023 · 1
Combine rules to find unknown angle

Going deeper

syllabusOfficial IB AA SL SyllabusFull unit learning outcomes

What's Next

Mastering the sine and cosine rules is a critical foundation for all further trigonometry topics in IB AA SL, including trigonometric functions, identities, equations, and 3D geometry applications. These rules appear frequently in both paper 1 and paper 2 questions, often as part of multi-step problems that combine multiple concepts. Once you are confident with these rules, you can move on to applications in 3D geometry, or continue to explore trigonometric identities and equations.