Edexcel IGCSE Maths 复习笔记 4.10.1 Sine & Cosine Rules, Area of a Triangle - Basics

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 RuleCosine 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

  • 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:

7.19.1-Sine-Cosine-Rules-Area-of-Triangle-basic-RN-1

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’:

7.19.1-Sine-Cosine-Rules-Area-of-Triangle-basic-RN-2

  • 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:

7.19.1-Sine-Cosine-Rules-Area-of-Triangle-basic-RN-3

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

7.19.1-Sine-Cosine-Rules-Area-of-Triangle-basic-Worked-Example-1General-Triangle-with-values7.19.1-Sine-Cosine-Rules-Area-of-Triangle-basic-Worked-Example-2

Worked Example

7.19.1-Sine-Cosine-Rules-Area-of-Triangle-basic-Worked-Example-3

转载自savemyexam

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