The unit circle and angles

The unit circle

The unit circle is a circle of radius 1 centred at the origin of an orthonormal coordinate system. For any angle x (measured in radians, oriented positively in an anti-clockwise direction), a point M on the circle is associated with it. The co-ordinates of M are exactly (cos(x), sin(x)). Thus, cos(x) is the x-co-ordinate and sin(x) is the y-co-ordinate of point M.

The preferred unit of measurement in trigonometry is the radian: a full turn measures 2π radians (i.e. 360 degrees), and a half-turn measures π radians (180 degrees).

Notable angles

Angle (rad)	0	π/6	π/4	π/3	π/2
cos(x)	1	√3/2	√2/2	1/2	0
sin(x)	0	1/2	√2/2	√3/2	1

Example: for x = pi/4 (45 degrees), cos(x) = sin(x) = sqrt(2)/2, which corresponds to an isosceles right-angled triangle.

Common pitfall

Never confuse degrees and radians in a formula, otherwise a misconfigured calculator will give an incorrect result. To convert: angle_rad = angle_deg * pi/180. Always check the mode (DEG or RAD) before performing a calculation.