Sine rule, cosine rule for non-right triangles

IB Mathematics AI SL· 5 min read

1. The Sine Rule★★☆☆☆⏱ 15 min

📘 Definition

Sine Rule

For triangle $A B C$ , with side $a$ opposite angle $A$ , $b$ opposite $B$ , $c$ opposite $C$ :

Relates the sides of any triangle to the sines of their opposite angles. It is used when you have a known side-angle opposite pair to find an unknown.

Example:

Find an unknown side when you know two angles and one side

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

📐 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 and unknown values:
2
$\frac{a}{\sin A} = \frac{b}{\sin B}$
3
Substitute the known values into the formula:
4
$\frac{8}{\sin 40^\circ} = \frac{b}{\sin 60^\circ}$
5
Rearrange the formula to isolate $b$ :
6
$b = \frac{8 \sin 60^\circ}{\sin 40^\circ}$
7
Calculate the value of $b$ , rounding to 3 significant figures:
8
$b \approx \frac{8 \times 0.8660}{0.6428} \approx 10.8 \text{ cm}$

2. The Cosine Rule★★★☆☆⏱ 20 min

📘 Definition

Cosine Rule

For triangle $A B C$ , with side $a$ opposite angle $A$ :

Relates three sides and one angle of any triangle, and works for all triangles, right or non-right. It can be rearranged to find an angle when all three sides are known.

Example:

Find an unknown side when you know two sides and the included angle

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

To find an unknown angle when all three sides are known, rearrange the cosine rule to isolate the cosine term:

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

📐 Worked Example

In triangle $X Y Z$ , $X Y = 7$ cm, $X Z = 9$ cm, and the angle at $X$ is $3 5^{\circ}$ . Find the length of $Y Z$ .

1
Label the triangle: $Y Z$ is opposite angle $X$ , so let $a = Y Z$ , $b = 9$ , $c = 7$ , $A = 3 5^{\circ}$
2
Apply the cosine rule for side $a$ :
3
$a^2 = 9^2 + 7^2 - 2(9)(7) \cos 35^\circ$
4
Calculate the right-hand side:
5
$a^2 = 81 + 49 - 126 \times 0.8192 \approx 26.78$
6
Take the square root to find $a$ , rounded to 2 decimal places:
7
$a = \sqrt{26.78} \approx 5.17 \text{ cm}$

3. Choosing the Correct Rule★★★☆☆⏱ 20 min

Choosing the right rule quickly saves time in exams and avoids unnecessary mistakes. The table below summarizes when to use each rule based on the information you are given:

Given Information	Rule to Use
Two angles + one side (AAS/ASA)	Sine Rule
Two sides + non-included angle (SSA)	Sine Rule (check ambiguous case)
Two sides + included angle (SAS)	Cosine Rule (find side)
Three sides (SSS)	Cosine Rule (find angle)

📐 Worked Example

A triangle has sides 5 cm, 7 cm, and 8 cm. Find the size of the largest angle.

1
The largest angle is opposite the longest side, so let $a = 8$ cm (opposite angle $A$ , the angle we need to find)
2
We have all three sides, so use the rearranged cosine rule for angles:
3
$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
4
Substitute the values $b = 5$ , $c = 7$ , $a = 8$ :
5
$\cos A = \frac{5^2 + 7^2 - 8^2}{2(5)(7)} = \frac{25 + 49 - 64}{70} = \frac{10}{70} \approx 0.1429$
6
Calculate the angle using inverse cosine:
7
$A = \arccos(0.1429) \approx 81.8^\circ$

Exam tip:

Always draw and label your triangle before starting any calculations to avoid mixing up opposite sides and angles.

4. Common Pitfalls

Wrong move:

Forgetting to check the ambiguous case for SSA sine rule problems

Why:

Two angles between 0° and 180° can have the same sine value, leading to two valid triangles

Correct move:

Calculate the second possible angle as $18 0^{\circ} - calculated angle$ , check if the sum of angles is less than 180°, and include both solutions if valid

Wrong move:

Mixing up opposite sides and angles when applying the rules

Why:

Sine and cosine rules always pair a side with the angle opposite it; swapping these gives incorrect results

Correct move:

Label the triangle clearly with side $a$ opposite angle $A$ before substituting any values into formulas

Wrong move:

Using Pythagoras' theorem on a non-right triangle

Why:

Pythagoras' theorem only holds for right-angled triangles, it will always give an incorrect value for non-right triangles

Correct move:

Always use the sine or cosine rule for any triangle that is not confirmed to be right-angled

Wrong move:

Using sine rule to find an angle when given three sides (SSS)

Why:

Sine rule requires a known side-angle opposite pair, which you do not have when all angles are unknown

Correct move:

Use the rearranged cosine rule to find the first angle (usually the largest opposite the longest side)

5. Quick Reference Cheatsheet

Given Information / Rule	Formula / Notes
AAS/ASA (2 angles + 1 side)	Sine Rule
SSA (2 sides + non-included angle)	Sine Rule, check ambiguous case
SAS (2 sides + included angle)	Cosine Rule (find side)
SSS (3 sides)	Cosine Rule (find angle)
Sine Rule	$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$
Cosine Rule (find side)	$a^{2} = b^{2} + c^{2} - 2 b c cos A$
Cosine Rule (find angle)	$cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$

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.

2022 · 1
Find unknown side in non-right triangle
2023 · 2
Calculate unknown angle with three sides

Going deeper

Formula sheetIB AI SL Formula Booklet Trigonometry SectionSine and cosine rules are provided

What's Next

Now that you can apply sine and cosine rules to non-right triangles, you have the foundational skills for solving most triangle-based problems in IB AI SL. These rules are regularly used for bearings, navigation, area calculations, and real-world modelling problems that appear in both paper 1 and paper 2. Mastery of choosing the correct rule quickly is critical for maximizing marks, as these questions are often structured to test your ability to identify which tool to use. Next, you will extend these skills to find the area of non-right triangles, then apply all triangle concepts to solve practical bearings problems.