Non-right-angled triangles

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Use the sine and cosine rules in calculations involving lengths and angles for any triangle
  2. Use the formula area of triangle = ½ ab sin C
  3. Solve problems involving obtuse angles and the ambiguous case

Introduction

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When you have a triangle that doesn't contain a right angle (90°), you can't use the basic trigonometry methods like SOH CAH TOA or Pythagoras' theorem. Instead, you need special rules called the sine rule and the cosine rule. These work for any triangle, whether the angles are acute (less than 90°), obtuse (between 90° and 180°), or even if one angle is exactly 90°.

Deciding Which Rule to Use

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Before solving any triangle problem, you need to choose the correct rule. Here's how to decide:

Step 1: Is the triangle right-angled?

  • If YES → use SOH CAH TOA or Pythagoras
  • If NO → continue to Step 2

Step 2: Do you have a complete angle-side pair?

An angle-side pair means an angle and the side directly opposite to it.

  • If YES (you have at least one complete pair) → use the Sine Rule
  • If NO (you don't have any complete pairs) → use the Cosine Rule

Step 3: Are you finding the area?

  • If you know two sides and the angle between them → use the Area Formula: Area = ½ ab sin C

Triangle Labeling Convention

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Before using any formula, you must label your triangle correctly:

  • Use capital letters (A, B, C) for the angles
  • Use lowercase letters (a, b, c) for the sides
  • Each side is labeled with the lowercase version of the angle opposite to it
    • Side a is opposite angle A
    • Side b is opposite angle B
    • Side c is opposite angle C

This labeling system makes the formulas work correctly.

The Sine Rule

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What is the Sine Rule?

The sine rule connects the sides and angles of any triangle. It states:

a/sin A = b/sin B = c/sin C

Or, you can flip it upside down:

sin A/a = sin B/b = sin C/c

When to use the Sine Rule

Use the sine rule when you have:

  • One complete angle-side pair (an angle and its opposite side)
  • Either another angle OR another side that you need to find

Finding a Missing Side

When finding a missing length, use the formula with sides on top:

a/sin A = b/sin B = c/sin C

Example: In triangle ABC, angle A = 30°, side a = 15 cm, and angle C = 40°. Find side c.

Solution:

a/sin A = c/sin C

15/sin 30° = c/sin 40°

c = (15 × sin 40°)/sin 30°

c = (15 × 0.6428)/0.5

c ≈ 19.3 cm

Finding a Missing Angle

When finding a missing angle, use the formula with angles on top:

sin A/a = sin B/b = sin C/c

Example: In a triangle, an angle of 30° is opposite a side of 12 cm. Find the angle x opposite a side of 18 cm.

Solution:

sin 30°/12 = sin x/18

sin x = (18 × sin 30°)/12

sin x = (18 × 0.5)/12

sin x = 0.75

x = sin⁻¹(0.75)

x ≈ 48.6°

Important note: The sin⁻¹ button (inverse sine) is only used when finding an angle, never when finding a side.

The Ambiguous Case

Sometimes when using the sine rule to find an angle, there can be two possible answers. This is called the ambiguous case.

Here's why this happens:

  • When you use sin⁻¹ on your calculator, it always gives you an acute angle (less than 90°)
  • But there might be an obtuse angle (between 90° and 180°) that has the same sine value

The rule for obtuse angles:

If the angle you're finding should be obtuse (you can tell from the diagram), use:

Obtuse angle = 180° - acute angle

Example: A triangle has sides 14 cm and 15 cm, with a 30° angle opposite the 14 cm side. Find the obtuse angle x opposite the 15 cm side.

Solution:

sin 30°/14 = sin x/15

sin x = (15 × sin 30°)/14

sin x = 0.5357

x = sin⁻¹(0.5357) = 32.4° (This is the acute answer)

Since the question states x is obtuse:
x = 180° - 32.4° = 147.6°

How to recognize the ambiguous case:

  • You're using the sine rule to find an angle
  • The diagram shows (or the question states) that the angle should be obtuse
  • Your calculator gives you an acute answer
  • You must subtract from 180°

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