Is Sin^2(a) Equal to Sin^2(b) + Sin^2(c) Only in a Right Triangle at A?

[Andrax]

Homework Statement

prove that sin^2(a)=sin^2(b)+sin^2(c) if and only if ABC is a right triangle in A

i worked really hard on this one I'm really confused why i didn' get the answer

Homework Equations

The Attempt at a Solution

a+b+c=pi
tried turning everythng to cos 2x didn't helpi really couldn't do this one, can't use complex numbers by the way...please help

 

[Snark1994]

Well, $a+b+c = \pi$ is going to be true of all triangles, so that's not necessarily going to be much help.

What does ABC being a right-angled triangle in A imply for the value of a? What does that imply for the value of b+c? What does that imply for the value of $\sin^2 a$?

 

[Andrax]

Well, $a+b+c = \pi$ is going to be true of all triangles, so that's not necessarily going to be much help.

What does ABC being a right-angled triangle in A imply for the value of a? What does that imply for the value of b+c? What does that imply for the value of $\sin^2 a$?

i tried everything you said also transforming $\sin^2 a$ won't help me since i want to keep sin a

 

[Snark1994]

transforming $\sin^2 a$ won't help me since i want to keep sin a

Transforming $\sin^2 a$ will help you. In fact, the easiest way I can see of proving the result from right-to-left (remember for an "if and only if" you need to prove it both ways) is to show that $\sin^2 a = k = \sin^2 b + \sin^2 c$ for some specific number k whose value you'll have to work out.

Like I asked, if the triangle is right-angled in A, what can you say about the value of 'a'? You can be very specific!

 

[SammyS]

$\sin((\pi/2) - \theta)=\cos(\theta)$

Also, for this to be true, a must be the right angle of the right triangle.

 

[Andrax]

$\sin((\pi/2) - \theta)=\cos(\theta)$

Also, for this to be true, a must be the right angle of the right triangle.

solved using Pythagoras's theorem --> let abc be a triangle wehave sin a^2 = AB^2 etc..
thank everyone..