# Sine rule, cosine rule for non-right triangles

> IB Mathematics AI SL · Unit 3: Geometry and Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u3-sine-rule-cosine-rule-for/

This topic covers how to find unknown sides and angles in non-right triangles using the sine and cosine rules. You will learn when to apply each rule to solve common triangle problems that appear frequently in IB exams.

**Prerequisites:** [Right-angled triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-right-angled-triangle-trig/); [Basic triangle angle properties](https://www.owlsprep.com/study/ib-math-ai-sl-u3-triangle-properties/)

## Learning objectives

- Apply the sine rule to find unknown sides and angles in non-right triangles
- Apply the cosine rule to find unknown sides and angles in non-right triangles
- Identify which rule to use for a given set of known triangle information
- Recognize and handle the ambiguous case of the sine rule

## The Sine Rule

**Sine Rule** — 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.

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

*Example:* Find an unknown side when you know two angles and one side

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

> **info**
>
> When given two sides and a non-included angle (SSA), you must check for the ambiguous case: two valid triangles may be possible.

**Worked example:** In triangle $ABC$, $A = 40^\circ$, $B = 60^\circ$, $a = 8$ 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. $$b \approx \frac{8 \times 0.8660}{0.6428} \approx 10.8 \text{ cm}$$

## The Cosine Rule

**Cosine Rule** — 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.

*Notation:* For triangle $ABC$, with side $a$ opposite angle $A$:

*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 $XYZ$, $XY = 7$ cm, $XZ = 9$ cm, and the angle at $X$ is $35^\circ$. Find the length of $YZ$.

1. Label the triangle: $YZ$ is opposite angle $X$, so let $a = YZ$, $b = 9$, $c = 7$, $A = 35^\circ$
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. $$a = \sqrt{26.78} \approx 5.17 \text{ cm}$$

## Choosing the Correct Rule

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.

## Common pitfalls

- **Wrong:** Forgetting to check the ambiguous case for SSA sine rule problems
  - Why it fails: Two angles between 0° and 180° can have the same sine value, leading to two valid triangles
  - Correct: Calculate the second possible angle as $180^\circ - \text{calculated angle}$, check if the sum of angles is less than 180°, and include both solutions if valid
- **Wrong:** Mixing up opposite sides and angles when applying the rules
  - Why it fails: Sine and cosine rules always pair a side with the angle opposite it; swapping these gives incorrect results
  - Correct: Label the triangle clearly with side $a$ opposite angle $A$ before substituting any values into formulas
- **Wrong:** Using Pythagoras' theorem on a non-right triangle
  - Why it fails: Pythagoras' theorem only holds for right-angled triangles, it will always give an incorrect value for non-right triangles
  - Correct: Always use the sine or cosine rule for any triangle that is not confirmed to be right-angled
- **Wrong:** Using sine rule to find an angle when given three sides (SSS)
  - Why it fails: Sine rule requires a known side-angle opposite pair, which you do not have when all angles are unknown
  - Correct: Use the rearranged cosine rule to find the first angle (usually the largest opposite the longest side)

## 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 | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ |
| Cosine Rule (find side) | $a^2 = b^2 + c^2 - 2bc \cos A$ |
| Cosine Rule (find angle) | $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ |

## 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.

- [Bearings and trigonometric applications](https://www.owlsprep.com/study/ib-math-ai-sl-u3-bearings-and-trigonometric-applications/)
- [Sinusoidal functions: amplitude, period, translation](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sinusoidal-functions-amplitude-period-translation/)
- [Voronoi diagrams and closest site problems](https://www.owlsprep.com/study/ib-math-ai-sl-u3-voronoi-diagrams-and-closest-site/)

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