When solving a trigonometric equation, explain how you know how many solutions there will be

The trig functions sine, cosine, tangent, etc., are cyclic. For example, if sin(x)=1/2, x=30 degrees is only one solution out of an infinite number because x=150, 390, 510, 750, etc., are also solutions. If sin(30x)=1/2, then x=5, 13, 17, 25, etc. Unless there is a specific restraint in the trig equation, there will always be an infinite number of solutions. For example, x=sin(x) has only solution, x=0. Another example is x=2sin(x), which has only three solutions. Sometimes a question will state the limits of the answer. For example, find x between the limits 0 to 360, or 0 to 2(pi), depending on whether x is measured in radians or degrees. In this range there are usually two solutions.

Graphically it is easier to tell how many solutions there will be. For example, x=2sin(x) can be represented by two functions plotted together on the same graph: y=x and y=2sin(x). The line y=x intersects the curve y=2sin(x) at 3 points, so there must be 3 solutions to x=2sin(x). You can see this, because the sine curve has humps, like the Loch Ness monster, that peak at 2 and -2 all the way along the x axis in both directions, positive and negative forever. The line joins the origin to (1,1), (2,2), etc. on the positive side and (-1,-1), (-2,-2), etc. on the negative side, and it cuts the sine curve through the first negative hump and the first positive hump including the origin, making three points of intersection only, because the line just rises over the humps after that.