Edexcel IGCSE Maths

Revision Notes

4.10.1 Sine & Cosine Rules, Area of a Triangle - Basics

What are the sine & cosine rules, & the area of a triangle formula?

  • Remember, Pythagoras and SOHCAHTOA only work in right-angled triangles
  • The Sine Rule, Cosine Rule, and Area of a Triangle Formula allow us to answer triangle questions for ANY triangle

The sine rule, cosine rule, & area of a triangle formula

  • In discussing these formulas, we usually label our triangle like this:

General Triangle labelled, IGCSE & GCSE Maths revision notes

  • Note: lowercase letters for side lengths, capital letters for angles — and make sure an angle and the side opposite it have the same letter
  • The Sine Rule tells us that:

Sine, Cosine Rules, Area of Triangle - basic RN 1, downloadable IGCSE & GCSE Maths revision notes

sin 90° =1 – so if one of the angles is 90°, this becomes ‘SOH’ from SOHCAHTOA

  • The Cosine Rule tells us that:
    a2 = b2 + c2 – 2b cos A
    cos 90° = 0   so if A = 90°, this becomes Pythagoras’ Theorem
  • The Area of a Triangle Formula tells us that:
    Area = ½ ab sin C
    sin 90° =1   so if C=90°, this becomes Area = ½ × base × height
  • The Sine Rule can also be written ‘flipped over’:

Sine, Cosine Rules, Area of Triangle - basic RN 2, downloadable IGCSE & GCSE Maths revision notes

  • This is more useful when we are using the rule to find angles
  • These two versions of the Cosine Rule are also valid for the triangle above:

b2 = a2 + c2 – 2ac cos B

c2 = a2 + b2 – 2ab cos C

Note that it’s always the angle between the two sides in the final term

  • These two versions of the Area of a Triangle Formula are also valid for the triangle above:

Sine, Cosine Rules, Area of Triangle - basic RN 3, downloadable IGCSE & GCSE Maths revision notes

Note that it’s always the angle between the two sides on the right-hand side –you can always just relabel the sides and angles of your triangle, though, instead of worrying about alternate versions of the last two formulas

Worked Example

Sine, Cosine Rules, Area of Triangle - basic Worked Example 1, downloadable IGCSE & GCSE Maths revision notes

General Triangle with values, IGCSE & GCSE Maths revision notes

Sine, Cosine Rules, Area of Triangle - basic Worked Example 2, downloadable IGCSE & GCSE Maths revision notes

Sine, Cosine Rules, Area of Triangle - basic Worked Example 3, downloadable IGCSE & GCSE Maths revision notes

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