Edexcel IGCSE Maths 复习笔记 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:

• 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:

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 – 2bc cos A

  cos 90° = 0   so if A = 90°, this becomes Pythagoras’ Theorem

• The Area of a Triangle Formula tells us that:sin 90° =1   so if C=90°, this becomes Area = ½ × base × height

  Area = ½ ab sin C

• The Sine Rule can also be written ‘flipped over’:

• 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:

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