Unrolling the Sine Graph

Walk the point once round the unit circle and note its height every quarter turn: 0 at 0°, 1 at 90°, 0 at 180°, −1 at 270° and 0 again at 360°. Lay those heights out along a straight angle axis, join them with the heights the circle gives in between, and you have the graph of y = sin(x).

On the second lap the point passes the same heights in the same order, so the graph repeats: sin(x + 360°) = sin(x) for every x. A graph that repeats like this is periodic, and the length of one repeat, 360°, is its period.

To find where the graph reaches a height, find the points of the circle at that height, then add 360° for each further lap. Height 1/2 comes at 30° and 150°, so over two laps the graph meets the line y = 1/2 at 30°, 150°, 390° and 510°.