SEC2_GYM · Trigonometry · TRIG.UNIT_CIRCLE

Unit circle and general trigonometric functions

On the unit circle, the point at angle θ has coordinates (cos θ, sin θ).

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Intuition

Understand first, compress later.

Sine and cosine are not only triangle ratios: they are the vertical and horizontal projections of circular motion.

Switch representations

Unit circleθ = 30° = π/6

radius = 1cos θ (horizontal)sin θ (vertical)angle θpoint P(cos θ, sin θ)

Point P on the unit circle has coordinates (cos θ, sin θ) = (0.866, 0.500). The cosine is the horizontal leg and the sine the vertical one; since the radius is 1, Pythagoras gives cos²θ + sin²θ = 1. That is why both values always stay between −1 and 1.

Formal view

The unit circle has radius 1. For every angle θ, , and tan when cos .

Why does it work?

The circle definition extends trigonometry consistently to negative angles and angles greater than .