Solutions of the equation of the form $\sin(x)=\sin(y)$

Question

Why does the equation " $\sin(x)=\sin(y)$" have solutions $x=y+2k \pi$ and $x=(\pi-y)+2k\pi$ ? For instance: if $\sin(30)=\sin(150)$, then we say $30=150+2k\pi$ but how come $30=510=870=$…?

Answer 1

That's because $\sin 30 \neq \sin 150$. We do have $\sin 30^\circ = \sin 150^\circ$, though, but that's a totally different story. In other words, you're mixing radians and degrees.

The degree-ifyed statement is that if $\sin x = \sin y$ then we have $$ x = y + k\cdot 360^\circ \quad \text{or}\quad x = 180^\circ - y + k\cdot 360^\circ $$ "$30=510=870=$" is outright false, although it is true that $\sin 30^\circ = \sin 510^\circ$ and $\sin 30^\circ = \sin 870^\circ$, because $$ 510^\circ = 180^\circ - 30^\circ + 360^\circ $$ and $$ 870^\circ = 180^\circ - 30^\circ + 2\cdot 360^\circ $$ as per the above.

Answer 2

If $ \sin y = \sin x $ then for $n=-4,-3,-2,-1,0,1,2,3,4.. $

$$ y = n \,180^{\circ} + (-1)^ n \cdot x $$