The definition of sin(a) is sin(a) = y/r, y is the y-coordinate of a point in the terminal side of the angle a. r is the distance from the point to the origin and r^{2} = x^{2} + y^{2}.
The definition of cos(a) is cos(a) = x/r, x is the x-coordinate of a point in the terminal side of the angle a. When a point in the terminal side of the angle a lies in quadrant I, the coordinate of any point in
quadrant I has positive x and positive y. so sin(a) in quadrant I is positive and cos(a) in quadrant I is also positive.

When a point in the terminal side of an angle a lies in quadrant II, its y-coordinate is positive and x-coordinate is negative, so sin(a) in quadrant II is positive and cos(a) in quadrant II is negative.

When a point in the terminal side of an angle a lies in quadrant III, its y-coordinate is negative and x-coordinate is negative, so sin(a) in quadrant II is negative and cos(a) in quadrant II is negative.

When a point in the terminal side of an angle a lies in quadrant IV, its y-coordinate is negative and x-coordinate is positive, so sin(a) in quadrant II is negative and cos(a) in quadrant II is positive.