If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.!
then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the answer..!! But i want a complete proof!
[Math] If $\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1$ then find the value of $\sin(c)$
trigonometry
Best Answer
Consider the two unit vectors on $S^2$ with spherical polar coordinates $(a, 0)$ and $(b,\frac{\pi}{2}-c)$, its components in $\mathbb{R}^3$ are:
$$\begin{align}&(\sin a, 0, \cos a)\\ \text{ and }\quad&(\sin b \cos(\frac{\pi}{2}-c), \sin b\sin(\frac{\pi}{2}-c), \cos b) =(\sin b\sin c,\sin b\cos c,\cos b) \end{align}$$ The angle $\theta$ between them satisfies:
$$\cos\theta = \cos a\cos b + \sin a\sin b\sin c$$
Geometrically, $\cos\theta = 1$ if and only if these two unit vector coincides. This in turn implies $a = b$ and $c = \frac{\pi}{2}$. By sine law, the two sides opposites to angle $a$, $b$ have equal length. So the triangle is an right angled isosceles triangle and $\sin c = 1$.
Alternatively, one can use the identity
$$\begin{align} & (\sin a-\sin b \sin c)^2+(\sin b\cos c)^2 + (\cos a-\cos b)^2\\ = & 2 \left( 1 - ( \cos a\cos b + \sin a \sin b\sin c)\right) \end{align}$$
to conclude $\cos c = 0$ and hence $\sin c = 1$.