sin(x) = sin(x + pi/3)
I know this is simple but so am i... thanks in advance for any help
If $\displaystyle x$ is a solution then $\displaystyle \sin(x)$ and $\displaystyle \sin(x+\pi/3)$ must lie at the intersections of some horizontal line through the unit circle. Furthermore, the angle between these points (measured from the origin) must be $\displaystyle \pi/3$. But the other angles between the rightmost point and the positive x axis and the leftmost point and the negative x axis must be equal by symmetry, so call them $\displaystyle \theta$.
This leads to the equation $\displaystyle 2\theta+\pi/3=\pi$, since together the three angles make a straight line. This gives you one solution and then the other is easily found.
Visually, you could use the Unit Circle.
$\displaystyle sin(x)$ gives the y co-ordinate of a point on the Unit Circle,
hence the y-axis from $\displaystyle -1$ to $\displaystyle 1$ acts as an axis of symmetry.
$\displaystyle 0\le\ x\ \le \pi\Rightarrow\ sin(x)=sin\left(x+60^o\right)$
requires that $\displaystyle x$ and $\displaystyle x+60^0$ are $\displaystyle 30^o$ either side of the y-axis.
$\displaystyle x=90^o-30^o=60^o$
The exact same logic may be used underneath the x-axis for $\displaystyle \pi\le\ x\ \le\ 2\pi$
to obtain a 2nd solution.
Complete the analysis by adding multiples of 360 degrees to the two solutions.