From angles to numbers: radians and the unit circle

A wheel-rim point rotates through 60°. How do we find its coordinates? Sine and cosine convert rotation into coordinates, including after full turns. Use functions and powers and roots.

Measuring rotation

Start from the positive horizontal direction. Counterclockwise is positive; clockwise negative. A full turn returns the point but adds 360° to the angle. π is circumference divided by diameter. A radian is arc length s divided by radius R:

$\theta=\frac{s}{R},\quad s=R\theta;\qquad360^\circ=2\pi\ \mathrm{rad},\quad180^\circ=\pi\ \mathrm{rad}.$

At 1 radian the arc equals the radius. Radians are dimensionless, but rad distinguishes them from degrees. The arc formula requires θ in radians.

Rotation and coordinates

x²+y²=1

Exact circle and projections at θ=π/3, P=(1/2,√3/2). Equal scales, lengths in radius units. The θ arc starts rightward and runs counterclockwise.

Four functions from two coordinates

On a radius-1 circle centered at (0,0):

$P=(\cos\theta,\sin\theta);\qquad\tan\theta=\frac{\sin\theta}{\cos\theta},\quad\cot\theta=\frac{\cos\theta}{\sin\theta}.$

Sine is the vertical coordinate; cosine the horizontal. Both exist for all real angles and lie in [−1,1]. Both are positive in quadrant I; cos is negative in II, both negative in III, sin negative in IV.

Tangent is undefined at θ=π/2+kπ, cotangent at θ=kπ, integer k: the respective denominator is zero. OpenStax

The triangle connection

For an acute angle, radius projections form a right triangle: sin=opposite/hypotenuse, cos=adjacent/hypotenuse. In other quadrants coordinates can be negative; side lengths do not become negative.

$\theta=\pi/3:\quad P=(1/2,\sqrt3/2),\quad\tan\theta=\sqrt3,\quad\cot\theta=1/\sqrt3.$

For R=2 m, the point is (1,√3) m and the arc is 2π/3 m. Angles θ+2πk share sin/cos values. Check DEG/RAD before calculating: the bare number 60 does not specify a rotation.