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Trigonometry · 02 · Where triangles become waves · 8 min

The unit circle

Put your triangle inside a circle of radius 1 and let the angle keep growing — past 90°, past 180°, around and around. Sine and cosine break free of triangles and become something bigger: the coordinates of rotation itself.

Build the intuition

Sine and cosine are coordinates

Stand at angle θ on a circle of radius 1: your horizontal position is cos θ, your height is sin θ. That's the entire definition. The triangle ratios were this picture all along, restricted to the first quarter-turn.

(x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos\theta, \sin\theta)

Why values repeat and go negative

Walk past 90° and your horizontal position goes negative (you're left of center) — cosine negative, no mystery. Complete a full turn and everything repeats, forever. Periodicity isn't a rule to memorize; it's what walking in circles means.

Radians: the natural angle

A radian measures angle by distance walked along the unit circle's edge: a full turn is 2π because that's the circumference. Degrees are arbitrary (Babylonian bookkeeping); radians are the circle measuring itself. Calculus strongly prefers them.

360°=2π radians360° = 2\pi \text{ radians}

See it move

InteractiveThe circle that makes waves
0.9
At angle 0.9 rad: height sin θ = 0.78, horizontal cos θ = 0.62. Walk the full circle and the height traces one complete sine wave — triangles and waves are the same picture.

Push the angle through a full turn: heights repeat, signs flip in the left and lower quarters, and the wave records the journey.

A worked example

Read values straight off the circle

  1. What is cos 180°? Walk half a turn: you stand at the far left of the circle, at point (−1, 0).

  2. Coordinates answer instantly:

    cos⁡180°=−1,sin⁡180°=0\cos 180° = -1, \quad \sin 180° = 0

  3. No formula, no table — the picture is the calculation.

Out in the world

Everything that rotates

Engine cranks, turbines, clock hands, spinning hard drives, Ferris wheels: any point on anything rotating has position (r cos θ, r sin θ). Game developers and roboticists type this pair daily — it's rotation's home address.

Common confusion, cleared

“sin of an angle bigger than 90° makes no sense.”

On the circle it's perfectly defined — it's just your height at that point of the walk, which may be negative. Triangles were the training wheels.

“Radians are a harder version of degrees.”

They're the less arbitrary version: angle measured by arc length. Formulas get simpler in radians — that's why mathematics standardized on them.

Recap

  • On the unit circle, cos θ and sin θ are simply your (x, y) position.
  • Negative values and repetition fall out of the geometry of walking in circles.
  • Radians measure angle by arc length: full turn = 2π.