Coordinates of a point: cosine and sine

Coordinates of a point on the unit circle

An angle θ, measured from point A(1,0) in the clockwise direction, defines a unique point M on the unit circle. The coordinates of this point M are denoted by (cos(theta), sin(theta)).

Definitions

• The cosine of theta is the x-coordinate of point M.
• The sine of theta is the y-coordinate of point M.

Thus, the coordinates of M are (cos(theta), sin(theta)).

Immediate properties

As the circle has a radius of 1 and centre O, we always have: -1 <= cos(theta) <= 1 -1 <= sin(theta) <= 1

Example

For theta = 0 (point A itself): cos(0) = 1 and sin(0) = 0. For θ = π/2 (a quarter turn clockwise): the point lies at the top of the circle, at (0,1), so cos(π/2) = 0 and sin(π/2) = 1. For θ = π (half a turn): the point is at (-1,0), so cos(π) = -1 and sin(π) = 0.

Signs by quadrant

The circle is divided into 4 quadrants (quarters):

Quadrant	Angle theta	Sign of cos	Sign of sin
1	0 to pi/2	+	+
2	pi/2 to pi	-	+
3	pi to 3pi/2	-	-
4	3pi/2 to 2pi	+	-

Common pitfall

Do not confuse cosine (x-coordinate, horizontal) and sine (y-coordinate, vertical). Mnemonic: associate cos with the horizontal axis (like ‘x’) and sin with the vertical axis (like ‘y’).