# Non-right triangles: Law of Sines, Law of Cosines, area

> IB Mathematics Applications and Interpretation HL · Unit 3: Geometry and trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u3-non-right-triangles-law-of/

This module covers methods for solving all unknown sides, angles and area of any non-right triangle, including the Law of Sines, Law of Cosines, and the sine-based area formula. You will learn when to apply each rule for common exam problems.

**Prerequisites:** [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-hl-u3-right-triangle-trigonometry/); [Sine and cosine of obtuse angles](https://www.owlsprep.com/study/ib-math-ai-hl-u2-trigonometric-functions/)

## Learning objectives

- Apply the Law of Sines to find unknown sides and angles in non-right triangles
- Apply the Law of Cosines to find unknown sides and angles in non-right triangles
- Calculate the area of any non-right triangle using the $\frac{1}{2}ab\sin C$ formula
- Identify and solve the ambiguous SSA case for non-right triangles

## The Law of Sines

**Law of Sines** — For any triangle with sides $a, b, c$ opposite angles $A, B, C$ respectively, the ratio of each side to the sine of its opposite angle is constant.

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

*Example:* Used when you know two angles and one side, or two sides and a non-included angle (SSA).

**Worked example:** In triangle $ABC$, $A = 40^\circ$, $B = 60^\circ$, $a = 8$ cm. Find the length of side $b$.

1. Write the Law of Sines ratio for the known and unknown values:
2. $$\frac{a}{\sin A} = \frac{b}{\sin B}$$
3. Substitute the known values into the equation:
4. $$\frac{8}{\sin 40^\circ} = \frac{b}{\sin 60^\circ}$$
5. Rearrange to isolate $b$:
6. $$b = \frac{8 \times \sin 60^\circ}{\sin 40^\circ}$$
7. Calculate with a calculator:
8. $$b \approx 10.8 \text{ cm}$$

> **tip**
>
> Always confirm your calculator is set to degree mode for all triangle problems in IB exams, unless explicitly told otherwise.

*Calculator:* allowed

## The Law of Cosines

**Law of Cosines** — Relates the three sides of a triangle to one of its angles. It is a general case of Pythagoras' theorem, which only applies to right triangles.

*Notation:* c^2 = a^2 + b^2 - 2ab\cos C \quad \text{or} \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}

*Example:* Used when you know three sides, or two sides and the included angle between them.

**Worked example:** In triangle $ABC$, sides $a = 5$ cm, $b = 7$ cm, included angle $C = 45^\circ$. Find the length of side $c$.

1. Select the form for finding an unknown side:
2. $$c^2 = a^2 + b^2 - 2ab\cos C$$
3. Substitute the known values:
4. $$c^2 = 5^2 + 7^2 - 2(5)(7)\cos 45^\circ$$
5. Calculate each term:
6. $$c^2 = 25 + 49 - 70(0.7071) \approx 24.50$$
7. Take the positive square root (side length cannot be negative):
8. $$c \approx \sqrt{24.50} \approx 4.95 \text{ cm}$$

> **note**
>
> When finding an unknown angle from three sides, use the rearranged $\cos C = ...$ form to simplify your calculation.

*Calculator:* allowed

## Area of a Non-Right Triangle

**Area of Any Triangle** — The area of any triangle is half the product of two sides multiplied by the sine of the included angle between them.

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

*Example:* Works for all triangles, right or non-right, when two sides and the included angle are known.

**Worked example:** Find the area of triangle $ABC$ with $a = 10$ m, $b = 14$ m, included angle $C = 30^\circ$.

1. Substitute into the area formula:
2. $$\text{Area} = \frac{1}{2}(10)(14)\sin 30^\circ$$
3. Simplify using $\sin 30^\circ = 0.5$:
4. $$\text{Area} = 70 \times 0.5 = 35 \text{ m}^2$$

If you know all three sides, first use the Law of Cosines to find any included angle, then apply this area formula. If you know two angles and one side, find the third angle, use the Law of Sines to find a second side, then calculate area.

*Calculator:* allowed

## Choosing the Right Rule & The Ambiguous Case

Knowing which rule to use saves time on exams and avoids unnecessary calculation. Use this guide:

| Given information | Recommended rule |
| --- | --- |
| Two angles + any side | Law of Sines |
| Two sides + non-included angle (SSA) | Law of Sines (check ambiguous case) |
| Two sides + included angle (SAS) | Law of Cosines + Area formula |
| Three sides (SSS) | Law of Cosines |

For SSA (two sides, non-included angle), there can be 0, 1, or 2 valid triangles. This is the ambiguous case, a common exam topic.

**Worked example:** Given $a = 7$, $b = 9$, $A = 25^\circ$, how many valid triangles exist?

1. Use Law of Sines to solve for $\sin B$:
2. $$\sin B = \frac{b \sin A}{a} = \frac{9 \sin 25^\circ}{7} \approx 0.543$$
3. Two angles between $0^\circ$ and $180^\circ$ have this sine value:
4. $$B_1 \approx 32.9^\circ, \quad B_2 = 180^\circ - 32.9^\circ = 147.1^\circ$$
5. Check if $B_2$ forms a valid triangle:
6. $$147.1^\circ + 25^\circ = 172.1^\circ < 180^\circ$$
7. Conclusion: Both angles are valid, so 2 distinct triangles exist.

> **tip**
>
> If the sum of the second angle and the known angle is less than 180°, two triangles exist. Always check this for SSA problems!

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using Law of Sines for two sides and an included angle (SAS)
  - Why it fails: Law of Sines requires the known angle to be opposite a known side, which is not true for SAS
  - Correct: Use Law of Cosines for all SAS problems
- **Wrong:** Forgetting to check the second angle in ambiguous SSA cases
  - Why it fails: IB exams explicitly test recognition of two valid triangles, so you will lose marks for missing the second solution
  - Correct: Always calculate the supplementary angle and check if it forms a valid triangle
- **Wrong:** Using radian mode on a calculator for triangle problems
  - Why it fails: All IB triangle problems use degrees unless stated otherwise, leading to drastically wrong results
  - Correct: Confirm your calculator is in degree mode before starting any trigonometry problem
- **Wrong:** Using the non-included angle in the $\frac{1}{2}ab\sin C$ area formula
  - Why it fails: The formula only works when the angle is between the two sides you use
  - Correct: Confirm your angle is included between the two sides, or find the included angle first with Law of Sines/Cosines
- **Wrong:** Keeping the negative square root when solving for a side with Law of Cosines
  - Why it fails: Side length is always positive, so including the negative root is an error
  - Correct: Only take the positive square root when calculating side length

## Cheatsheet

| Rule | Formula | When to use |
| --- | --- | --- |
| Law of Sines | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ | 2 angles + 1 side, SSA |
| Law of Cosines (side) | $c^2 = a^2 + b^2 - 2ab\cos C$ | SAS, find third side |
| Law of Cosines (angle) | $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$ | SSS, find angle |
| Area of Triangle | $\frac{1}{2}ab\sin C$ | SAS, find area |

## What's next

Mastering non-right triangle trigonometry is foundational for many higher topics in IB AI HL, including 2D navigation with bearings, and modelling real-world scenarios involving distances and angles. The rules you learned here extend directly to problems involving triangles embedded in 3D shapes, where you will break complex 3D problems into multiple 2D non-right triangles to solve for unknown heights or distances. These methods also underpin work with area of compound shapes and trigonometric modelling, which appear regularly on both Paper 1 and Paper 2 exams.

- [Unit circle, radian measure, trigonometric identities](https://www.owlsprep.com/study/ib-math-ai-hl-u3-unit-circle-radian-measure-trigonometric/)
- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-ai-hl-u3-graphs-of-trigonometric-functions/)
- [Solving trigonometric equations](https://www.owlsprep.com/study/ib-math-ai-hl-u3-solving-trigonometric-equations/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-hl-u3-non-right-triangles-law-of/