I am supposed to find the value of $\sin^2\alpha+\sin^2\beta+\sin^2\gamma$ and I have been provided with the information that $\sin \alpha+\sin \beta+\sin\gamma=0=\cos\alpha+\cos\beta+\cos\gamma$.
I tried to approach this using vectors. We can consider three unit vectors that add up to $0$. Unit vectors because the coefficients of the $\sin$ and $\cos$ terms are $1$. For the sake of simplicity, let one of the vectors $\overline{a}$ be along the $x$-axis. Let the angles between $\overline{b}$ and $\overline{c}$ be $\alpha$, between $\overline{a}$ and $\overline{b}$ be $\gamma$ and between $\overline{a}$ and $\overline{c}$ be $\beta$. Then we have:
$$\begin{aligned}\overline{a}&=\left<1,0\right>\\ \overline{b}&=\left<-\cos\gamma, -\sin\gamma\right>\\ \overline{c}&=\left<-\cos\beta, \sin\beta\right>\end{aligned}$$
$$\begin{aligned}\cos\gamma+\cos\beta &=1\\ \sin\beta&=\sin\gamma\end{aligned}$$
Now, $\cos \gamma$ and $\cos\beta$ must have the same sign. So we get $\sin\alpha=-\sqrt{3}/2$, $\sin\beta=\sqrt{3}/2$ and $\sin\gamma=\sqrt{3}/2$. This contradicts with the answer key provided according to which $\sum_{cyc}\sin^2\alpha=3/2$. What am I doing wrong?
This was the picture I had in mind with $\overline{a}$ aligned with the horizontal.
Best Answer
In the case you are dealing with, the three vectors are $(1,0)$, $(-1/2,\sqrt3/2)$ and $(-1/2,-\sqrt3/2)$. The sum of the squares of the $y$-coordinates is $$0^2+\left(\frac{\sqrt3}2\right)^2+\left(-\frac{\sqrt3}2\right)^2=\frac32.$$
The general solution though, has vectors $(\cos\alpha,\sin\alpha)$, $(\cos(\alpha+2\pi/3),\sin(\alpha+2\pi/3))$ $(\cos(\alpha-2\pi/3),\sin(\alpha-2\pi/3))$ so you need to prove the identity $$\sin^2\alpha+\sin^2\left(\alpha+\frac{2\pi}3\right) +\sin^2\left(\alpha-\frac{2\pi}3\right)=\frac32.$$