what is the method for determining the answers for this simple trig equation? how do you determine the answers for $\sin 2 \theta =0$ within the range $0\leqslant\theta\leqslant2\pi $ ?
The answers I get from looking at a table of $\sin \theta$ values are
$0$, $\pi$ and $2 \pi$ 
while the answers provided by a specimen paper from the examination body are 
$0$, $\pi/2$, $\pi$, and $3\pi/2$
What is the correct way to determine the answers and are my answers wrong?
 A: You have forgotten that you are solving:
$$sin(2\theta)$$ not $$sin(\theta)$$
You have found that:
$$2\theta=0$$
$$2\theta=\pi$$
$$2\theta=2\pi$$
Therefore:
$$\theta=0$$
$$\theta=\pi/2$$
$$\theta=\pi$$
However, there are more answers in the range given. These are:
$$\frac{3\pi}{2}$$ and $$2\pi$$
The answer book has forgotten about $$2\pi$$
However, it is an answer because the range is less than or equal to.
Hope this helps!
A: You should really look in the range $[0,4\pi]$ for values of $\sin(\theta)=0$ if you want the values of $\sin(2\theta)=0$ to be in the range $[0,2\pi]$.
This is essentially because if we found values in $[0,4\pi]$ which solve $$\sin(\theta)=0,$$ then since this is the same as $$\sin\left(2\cdot\left(\frac{\theta}{2}\right)\right)=0,$$
We can use the same values of $\theta$ found except divided by two, which now puts them in the range $[0,2\pi]$.
Just to make this more explicit, the solutions in $[0,4\pi]$ are $$0,\pi,2\pi,3\pi,4\pi,$$ and now divide these by two to give:
$$0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi.$$
