# Sine rule, cosine rule and area of triangles

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u3-sine-rule-cosine-rule-and/

This module extends right-angle trigonometry to all oblique (non-right) triangles, teaching how to find missing sides, angles, and calculate area. We cover when to use each rule and common exam traps to avoid.

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

## Learning objectives

- Apply the sine rule to find missing sides and angles in any triangle
- Apply the cosine rule to find missing sides and angles in any triangle
- Calculate the area of any triangle using $\frac{1}{2}ab\sin C$
- Identify which rule to use for different triangle problem types

## The Sine Rule

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

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

*Example:* If $a=5$, $A=30^\circ$, $B=60^\circ$, 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 $ABC$, $A = 40^\circ$, $B = 60^\circ$, $a = 8$ 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.

## Ambiguous Case of the Sine Rule

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.

**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 $ABC$ with $a = 7$, $b = 9$, $A = 35^\circ$, 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 $\boldsymbol{\sin B}$:

   $$\sin B = \frac{9 \times \sin 35^\circ}{7} \approx 0.738$$
3. Use the identity $\boldsymbol{\sin \theta = \sin (180^\circ - \theta)}$ 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: $35^\circ + 132.4^\circ = 167.4^\circ < 180^\circ$, so both are valid.

> **warning**
>
> Always check for a second solution in SSA sine rule problems unless the question specifies the triangle is acute/obtuse.

## The Cosine Rule

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

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

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

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

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 $ABC$, $b = 4$ cm, $c = 6$ cm, included angle $A = 50^\circ$. 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 $\boldsymbol{\cos A}$ tells you if the angle is acute ($\cos A > 0$) or obtuse ($\cos A < 0$), no extra checks needed.

## Area of Any Triangle

**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 $XYZ$ where $XY = 5$ cm, $XZ = 7$ cm, angle at $X$ is $30^\circ$.

1. Identify the two sides and their included angle: $XY=5$, $XZ=7$, included angle $X=30^\circ$
2. Substitute into the area formula:

   $$Area = \frac{1}{2} \times 5 \times 7 \times \sin 30^\circ$$
3. Calculate, using $\boldsymbol{\sin 30^\circ = 0.5}$:

   $$Area = 8.75 \text{ cm}^2$$

## Common pitfalls

- **Wrong:** Mixing up sides and angles in sine rule, using $a/\sin B$ instead of $a/\sin A$
  - Why it fails: Sine rule requires each side divided by the sine of the angle opposite it, not adjacent.
  - Correct: Label sides with lowercase letters matching the uppercase angle opposite them before starting calculations.
- **Wrong:** Forgetting to check for a second solution in the ambiguous SSA case
  - Why it fails: Most students only calculate the acute solution, missing the valid obtuse solution and losing marks.
  - Correct: Always calculate $180^\circ - \arcsin(x)$ and check if total angles sum to less than $180^\circ$ to see if it is valid.
- **Wrong:** Using sine rule when given two sides and their included angle
  - Why it fails: You do not have a side-opposite-angle pair to apply the sine rule directly here.
  - Correct: Use cosine rule first to find the third side, then use sine rule if you need additional angles.
- **Wrong:** Applying $\frac{1}{2}ab\sin C$ with a non-included angle
  - Why it fails: The formula only works when the angle is between the two sides you are using.
  - Correct: Find the included angle first using sine or cosine rule before calculating area.
- **Wrong:** Rounding intermediate steps too early, leading to inaccurate final answers
  - Why it fails: Early rounding introduces cumulative error that can change the final answer enough to lose accuracy marks.
  - Correct: Keep full precision on your calculator until the final step, then round to the required number of significant figures.

## 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) | $\frac{1}{2}ab\sin C$ |

## 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.

- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-sl-u3-graphs-of-trigonometric-functions/)
- [Pythagorean and double angle trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-sl-u3-pythagorean-and-double-angle-trigonometric/)
- [Solving trigonometric equations](https://www.owlsprep.com/study/ib-math-aa-sl-u3-solving-trigonometric-equations/)

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