The signs of sine in the coordinate quadrants
Deriving the signs for the sine
By definition of the sine: The sine of angle α is the ordinate of point М (y of point М) on the trigonometric circle formed by the rotation of radius vector OM by angle α.
This definition follows from determining the sine through a triangle. The sine is the ratio of its opposite leg to the hypotenuse.
Opposite leg MD (equals y), hypotenuse OM (since the circle is a unit circle, OM = 1). Therefore,
i.e., the sine equals the ordinate of point M.