The Law of Cosines

The Law of Cosines

The Law of Cosines is the relationship between sides and angles in any triangle.

Introduction

Let's prove the theorem.
  • Draw a triangle ABC,then draw three squares sharing their sides with the triangle ABC.
  • Draw three lines through A,B,and C so that each line is perpendicular to the opposite side.
  • You can prove A1=A2, A3=A4, and A5=A6 by using the applet bellow.
  • Now you understand
    a^2=A2+A3=A1+A4=(c^2-A6)+(b^2-A5)
    =c^2+b^2-2*A6
  • Find the expression for the area A6.

The Law of Sines


The Law of Sines is the relationship between sides and angles in any triangle.


Introduction

Let's prove the theorem.

(Hints)

MENU1...If a sinB = b sinA, you can make a triangle.

MENU2...If a triangle exists, always a sinB = b sinA.

MENU3...Let R be the radius of the circumscribed circle of triangle ABC, then 2RsinC = c.