# 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.

Deriving the signs of the sine in coordinate quadrants

## Therefore, sines of the angles

• that end in quadrants I and II are positive;
• that end in quadrants III and IV are negative.

The signs of sine in the coordinate quadrants