find angle sine knowing all sides

Question

I know all the sides of an arbitrary triangle but not the angles, and I want to find the sine of any angle.

       A
       ^
     b/ \c
     /   \
   C/_____\B
       a

I know that $\cos(C)=\frac{a²+b²-c²}{2ab}$ (where side 'c' is opposite to angle 'C').
I want a similar one for sin
The only thing I got is $\sin(C) = \sqrt{1-\cos^2(C)}$ I want a simpler one .. does it exist?

Answer 1

you can find area of triangle . then you can find sin A by using the formulas below

Answer 2

We know that $c^2=a^2+b^2-2ab \cos{C}$, and you know $a$, $b$, and $c$, so plug it in and find out the $\cos$ of that angle. If you want to find the sine, just use any conversion formula, such as $\sin{x}^2+\cos{x}^2=1$ or any other conversion necessary. This is the simplest way. Darya's answer is just as valid, but just takes more steps and calculates unnecessary information.

Also I just want to say that its been an honor helping the President with his geometry work.

Answer 3

This seems to be the simplest form:

$\frac{sin A}{a}=\frac{sin B}{b}=\frac{sin C}{c}=\frac{1}{2 a b c } \sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}$