How to prove the law of cosines for all types of triangles?

[vortex193]

Homework Statement

a^2 = b^2 + c^2 - 2bc(cosA)

Confirm that this is true.

Hint: Drop a perpendicular from angle B to side b and use the two right triangles formed.

Homework Equations

The Attempt at a Solution

Honestly, I am looking at my diagram and simply cannot figure out what to do.

 

[fzero]

Can you use trigonometry to determine the length of the perpendicular that it's suggested that you draw? Is there a formula that would let you relate this to the other sides of the right triangles formed?

 

[vortex193]

I'm pretty sure that you can use trigonometry to determine the length of the perpendicular. But I only know that you have to show that this equation is true.

 

[fzero]

I'm pretty sure that you can use trigonometry to determine the length of the perpendicular. But I only know that you have to show that this equation is true.

The equation that you're trying to verify relates the squares of the length of the sides of the original triangle to each other. Are any of these sides shared with the right triangles that the hint tells you to form? Is there a chance that applying the Pythagorean theorem to these right triangles would provide useful information for this problem?

 

[zgozvrm]

Try drawing in the height of the triangle, then utilize the Pythagorean theorem. You should be able to prove this for obtuse, acute and right triangles.