The unit circle and radians

Picture a circle of radius 1, centred at the origin of the coordinate system: this is the unit circle. We always travel on it the same way: starting at point (1, 0), on the right, and turning counterclockwise, called the positive direction. This circle is about to become your map of the world of angles.

Now, a beautiful idea: take the number line, flexible like a thread, and wrap it around this circle. Every real number lands on a point of the circle. The number t lands at the point reached after an arc of length t from the start. This is how the radian is defined: an angle of 1 radian intercepts an arc of length 1 on the circle of radius 1. Angle and arc length become one and the same!

The full lap around the circle measures 2π (the perimeter of a circle of radius 1). A half turn is therefore π radians, which corresponds to 180 degrees. This equivalence, π rad = 180°, is the key to every conversion: to go from degrees to radians, multiply by π/180; the other way, multiply by 180/π. The famous angles quickly become familiar: 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6.

Why abandon degrees, which worked just fine? Because the radian is a natural measure: on a circle of radius r, an angle of t radians intercepts an arc of length exactly r × t. No conversion formula, geometry becomes direct computation. This simplicity is what will power all of trigonometry, then the analysis of the sine and cosine functions.

One last landmark: since a full lap is 2π, two numbers that differ by 2π (or 4π, 6π...) wrap onto the same point of the circle. The unit circle is an infinite clock: turn as much as you like, you keep passing through the same points. Keep this picture in mind, it lights up the whole chapter.