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,

Знаки тригонометрических функций

i.e., the sine equals the ordinate of point M.

The signs of sine in the coordinate quadrants

Deriving the signs of the sine in coordinate quadrants

Therefore, the sign of the sine is determined by the sign of the ordinate of (y):

  • the sines of the angles are positivefor the angles that end in the quadrant of the coordinate plane in which the points’ ordinates are positive.

  • the sines of the angles are negativefor the angles that end in the quadrant of the coordinate plane in which the points’ ordinates are negative.

Therefore, sines of the angles

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

Знаки синуса

The signs of sine in the coordinate quadrants