Unit Circle Reference

Keep this open while you code. Everything here follows from one fact: the point at angle θ on a circle of radius 1 is (cos θ, sin θ).

Conversions

Anchors worth memorising: 1 rad ≈ 57.3°, π ≈ 3.1416, 2π ≈ 6.2832.

Landmark angles

The first quadrant is the only part worth memorising. Everything below 90° repeats in the other three quadrants with signs flipped, which the next table gives you.

Quadrant signs

Counterclockwise from the positive x-axis, with +y upward:

Cosine is positive on the right half of the circle; sine is positive on the top half. That is easier to reconstruct than any mnemonic.

Bounds and identities

• −1 ≤ cos θ ≤ 1 and −1 ≤ sin θ ≤ 1, for every θ. A value outside this range is a bug.
• cos²θ + sin²θ = 1 — Pythagoras on a radius-1 circle. Useful as an assertion.
• cos(−θ) = cos θ, sin(−θ) = −sin θ — cosine is symmetric about the x-axis, sine flips.
• cos(θ + 2π) = cos θ — everything repeats every full turn, so angles are never “too big”.
• sin(θ + π/2) = cos θ — the same wave, shifted a quarter turn. Cosine leads, sine follows.

Recipes

Direction from an angle (a unit vector — pure direction, no speed):

dx, dy = math.cos(angle), math.sin(angle)

Move forward at a speed, frame-rate independent:

x = x + math.cos(angle) * speed * dt
y = y + math.sin(angle) * speed * dt

Orbit a centre at a radius:

x = cx + math.cos(angle) * radius
y = cy + math.sin(angle) * radius

Place n objects evenly around a full circle:

for i = 0, n - 1 do
  local a = i * (2 * math.pi) / n
  place(cx + math.cos(a) * radius, cy + math.sin(a) * radius)
end

Fan n projectiles across a total spread, centred on aim:

for i = 0, n - 1 do
  local a = aim + (i - (n - 1) / 2) * (spread / math.max(n - 1, 1))
  fire(math.cos(a), math.sin(a))
end

Traps checklist

When circular motion looks wrong, walk this list before debugging anything else:

1. Degrees where radians belong. sin/cos always take radians. Raylib’s drawing calls take degrees; LÖVE’s rotate takes radians. A 57× speed error is this.
2. y grows downward. On screen, increasing θ sweeps clockwise, not counterclockwise. The formula is unchanged; only your expectation is.
3. cos and sin swapped. Mirrors everything across the 45° diagonal. Looks plausible, reads as “the controls feel off”.
4. Speed baked into the direction. Keep (cos θ, sin θ) at length 1 and multiply by speed separately, or diagonal movement drifts faster than straight movement.
5. Angle never wrapped. Fine for sin/cos, which repeat every 2π — but it breaks the moment you start comparing or interpolating angles.