Proof of the cosine rule. Proving the cosine rule using Pythagoras and Trigonometry.
In this hub page I will show you how you can prove the cosine rule:
a² = b² + c² -2bcCosA
First of all draw a scalene triangle and name the vertices A,B and C. The capital letters represent the angles and the small letters represent the side lengths that are opposite these angles.
Now the trick is to split up triangle ABC into 2 right angled triangles by drawing a vertical line from vertex B down to the line AC. You can call this new line h (AD). You now have two right angled triangles (AND and BCD).
You can now introduce a new variable x, which will be the length of the line AD in the left right angled triangle. Therefore the length of CD in the other right angled triangle will be b – x.
Next, use Pythagoras Theorem in both triangles to write down two formulas that give h²:
h² = c² - x² (using triangle AND)
h² = a² - (b – x)² (using triangle BCD)
Now if you put these two formulas together you get:
c² - x² = a² - (b-x)²
The next thing you need to do is expand and simplify the double bracket on the right hand side of the formula (be careful with your negatives):
c² - x² = a² - [(b-x)(b-x)]
c² - x² = a² - [b² - 2bx +x²]
c² - x² = a² - b² + 2bx - x²
Now switching some of the terms from side to side, and cancelling out the x² on both sides you get:
a² = b² + c² - 2bx
All you need to do now is get rid of the letter x. This is done by using basic trigonometry in the triangle AND. Since CosѲ = adjacent/hypotenuse, then x can be expressed as:
x = cCosA
So the final step is to substitute x = cCosA into a² = b² + c² - 2bx:
a² = b² + c² - 2bcCosA (just change x to cCosA)
And this is your cosine rule proved.