Using the Cosine Law on Right and Oblique Triangles

[supernova1203]

Can the cosine law be used on right triangles as well as oblique triangles?

 

[dextercioby]

What do you think ?

 

[supernova1203]

I don't know, that's why i posted, but I am going to take a guess with... YES?

 

[HallsofIvy]

Of course, it can. Both the cosine law and sine law apply to all triangles. Of course, you don't really need it. Just use the definitions or sine and cosine to find lengths or angles.

The cosine law says $c^2= a^2+ b^2- 2ab cos(\theta)$.

If $\theta$ is a right angle, its cosine is 0 so that just gives the Pythagorean theorem. If it is one of the other angles, we can rewrite it as $a^2= b^2+ c^2- 2ac cos(\theta)$ with "c" still the length of the hypotenuse. Dividing by $c^2$, that becomes $sin^2(\theta)= cos^2(theta)+ 1- 2cos^2(\theta)$ so that $sin^2(\theta)- cos^2(\theta)= (1- cos^2(\theta))- cos^2(\theta)$ which is obviously true.

 

[mharten1]

Yes. But there's no point in using it on a right triangle, because the normal ratios apply.

It's only useful for oblique triangles because you can't use the normal trig functions.