# Non-right triangle trigonometry

> IB Mathematics AA HL · Geometry & Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u3-non-right-triangle-trigonometry/

This module covers trigonometric tools for solving any oblique (non-right) triangle. You will learn the sine rule, cosine rule, area formula, and how to handle the ambiguous sine rule case that is commonly tested in exams.

**Prerequisites:** [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-right-triangle-trigonometry/); Basic triangle angle sum properties

## Learning objectives

- Apply the sine rule to solve non-right triangles and identify the ambiguous case
- Apply the cosine rule to solve SSS and SAS triangle problems
- Calculate the area of any non-right triangle using the $\frac{1}{2}ab \sin C$ formula
- Select the correct trigonometric rule for a given problem setup

## The Sine Rule and Ambiguous Case

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

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

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

The sine rule is used when we know either (1) two angles and one side (AAS/ASA), or (2) two sides and a non-included angle (SSA). The second SSA case is called the *ambiguous case*, because two different triangles can satisfy the given measurements.

**Worked example:** Find all possible values of angle $B$ in triangle $ABC$ where $a = 10$ cm, $A = 30^\circ$, $b = 12$ cm

1. Apply the sine rule, rearrange to solve for $\sin B$:
2. $$\frac{10}{\sin 30^\circ} = \frac{12}{\sin B}$$
3. Calculate the left-hand side, since $\sin 30^\circ = 0.5$:
4. $$\sin B = \frac{12 \times 0.5}{10} = 0.6$$
5. The first solution from the calculator is the acute angle:
6. $$B_1 = \arcsin(0.6) \approx 36.9^\circ$$
7. The second possible solution is obtuse, since $\sin \theta = \sin(180^\circ - \theta)$:
8. $$B_2 = 180^\circ - 36.9^\circ = 143.1^\circ$$
9. Check validity: $143.1^\circ + 30^\circ = 173.1^\circ < 180^\circ$, so both solutions are valid.

> **tip**
>
> Always check for the second possible angle when using the sine rule for SSA cases. Exam questions regularly test this step.

## The Cosine Rule

**Cosine Rule** — Relates the lengths of sides to the cosine of an angle, with two common forms for finding sides or angles

*Notation:* Same notation for triangle $ABC$ as before

*Example:* Side form: $c^2 = a^2 + b^2 - 2ab\cos C$; Angle form: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

The cosine rule is used when we know either (1) three sides (SSS), or (2) two sides and the included angle (SAS). Unlike the sine rule, there is no ambiguous case here, because the cosine of an obtuse angle is negative, giving a unique solution.

**Worked example:** Find the length of side $c$ in triangle $ABC$ where $a = 5$ cm, $b = 7$ cm, and the included angle $C = 60^\circ$

1. Use the side form of the cosine rule:
2. $$c^2 = a^2 + b^2 - 2ab\cos C$$
3. Substitute the known values, $\cos 60^\circ = 0.5$:
4. $$c^2 = 5^2 + 7^2 - 2(5)(7)(0.5) = 25 + 49 - 35 = 39$$
5. Take the positive square root for side length:
6. $$c = \sqrt{39} \approx 6.25 \text{ cm (3 s.f.)}$$

**Check your understanding**

Test your understanding of when to use the cosine rule:

1. Which of the following cases requires the cosine rule?

   - Two angles and one side
   - Two sides and non-included angle
   - Two sides and included angle
   - All of the above

   *Answer:* Two sides and included angle

   *Why:* Correct. Only SAS or SSS cases require the cosine rule; other cases are solved faster with the sine rule.

## Area of a Non-Right Triangle

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

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

This formula works for all triangles, right and non-right. If you know all three sides but no angles, you can first use the cosine rule to find any included angle, then apply the area formula.

**Worked example:** Find the area of triangle $ABC$ with $a = 4$ cm, $b = 6$ cm, and included angle $C = 35^\circ$

1. Substitute into the area formula:
2. $$\text{Area} = \frac{1}{2} (4)(6) \sin 35^\circ$$
3. Calculate the result, rounding to 3 significant figures:
4. $$\text{Area} = 12 \times 0.5736 \approx 6.88 \text{ cm}^2$$

> **note**
>
> This area formula is very commonly used in multi-part exam questions that combine multiple trigonometric rules.

## Choosing the Correct Rule

**Comparing methods**

The rule you choose depends entirely on the information you are given. Here is a quick comparison:

- **Sine Rule** — Use for AAS, ASA, or SSA (two sides + non-included angle)
  - Pros: Simpler calculations than the cosine rule
  - Cons: Requires checking for ambiguous case in SSA

- **Cosine Rule** — Use for SAS (two sides + included angle) or SSS (three sides)
  - Pros: No ambiguous case, always gives a unique solution
  - Cons: More arithmetic steps, higher chance of substitution error

- **Area Formula** — Use to find area when you know two sides and included angle
  - Pros: Fast calculation once the angle is known
  - Cons: Cannot be used directly without an included angle

**Exam command terms**

Common exam command terms for this topic have specific expectations:

- **Solve the triangle** — Find all unknown side lengths and all unknown angles *(Always give answers to the required number of significant figures)*

- **Find all possible values** — Explicitly check for the ambiguous case and list all valid solutions

## Common pitfalls

- **Wrong:** Forgetting to check the second solution in SSA sine rule cases
  - Why it fails: Calculators only return the acute solution for arcsine, so you will miss the second valid obtuse solution
  - Correct: Always calculate $180^\circ - \arcsin(\sin B)$ and check if the sum with the known angle is less than $180^\circ$; add it to your answer if valid
- **Wrong:** Matching the wrong side to the wrong angle in the sine/cosine rule
  - Why it fails: Mislabeling the triangle leads to completely incorrect calculations
  - Correct: Always label the triangle clearly before starting, confirm side $a$ is always opposite angle $A$
- **Wrong:** Rounding intermediate values too early
  - Why it fails: Early rounding leads to inaccurate final answers that lose accuracy marks
  - Correct: Keep full calculator precision for all intermediate steps, only round your final answer
- **Wrong:** Using the sine rule for an included angle between two known sides
  - Why it fails: This adds unnecessary steps and increases the chance of error
  - Correct: Use the cosine rule directly for SAS cases to find the unknown side immediately
- **Wrong:** Wasting time calculating an unknown height for area
  - Why it fails: Many students try to find height to use $\frac{1}{2} \times \text{base} \times \text{height}$ when the simpler formula works
  - Correct: If you know two sides and the included angle, use $\frac{1}{2}ab \sin C$ directly

## Cheatsheet

| Given Information | Rule to Use |
| --- | --- |
| Two angles + one side | Sine Rule |
| Two sides + non-included angle | Sine Rule (check ambiguous case) |
| Two sides + included angle (find side) | Cosine Rule |
| Two sides + included angle (find area) | $\frac{1}{2}ab \sin C$ |
| Three sides (find angles) | Cosine Rule |

## What's next

Non-right triangle trigonometry is the foundation for all advanced trigonometric and vector topics in IB AA HL. You will next apply these rules to 3D geometry, solving for distances and angles between lines and planes. The cosine rule is also directly related to the vector dot product formula, which you will use extensively when working with vectors in 2D and 3D. Mastering the skill of choosing the right rule and checking for the ambiguous case will help you avoid common mistakes in these more complex topics later in the course.

- [Trigonometric Identities](https://www.owlsprep.com/study/ib-math-aa-hl-u2-trigonometric-identities/)
- [Trigonometric equations](https://www.owlsprep.com/study/ib-math-aa-hl-u3-trigonometric-equations/)
- [Compound and double angle identities](https://www.owlsprep.com/study/ib-math-aa-hl-u3-compound-and-double-angle-identities/)

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