There is no headache, once you realize that sinusoidal waves are symmetrical.
The minimum at (2?, -1) is horizontally 3/4ths the distance from the origin to the point at (P, 0).
In other words, if you divide the line segment from (0, 0) to (P, 0) into four equal pieces, 2? is 3 pieces away from the origin.
The second "hump" within any period of a sine wave is always centered horizontally three-quarters the way along the period. (You know that the distance P is the period, right?) So, we solve the following for P.
(3/4) P = 2?