Cheatsheet: Trig derivatives from geometry

The results

d/dx( sin(x) ) = cos(x)
d/dx( cos(x) ) = -sin(x)

Minus sign on cosine. Derived, not memorized.

The derivation (one picture)

1. A point at angle x on the unit circle sits at (cos x, sin x). Sine and cosine are its coordinates.
2. Radians make arc length equal angle, so nudging x by dx moves the point arc length dx: unit speed.
3. Moving on a circle, velocity is perpendicular to position (tangent), length 1, pointing 90° counterclockwise.
4. Rotating (a, b) by 90° CCW gives (-b, a). So velocity = (-sin x, cos x).
5. Velocity components are the coordinates’ rates of change:
   • horizontal coord cos x changes at -sin x -> d/dx(cos x) = -sin x
   • vertical coord sin x changes at cos x -> d/dx(sin x) = cos x

Sanity checks

x	sin / its slope cos	cos / its slope -sin
0	sin 0 = 0, slope cos 0 = 1 (climbing)	cos 0 = 1 peak, slope 0 (flat)
π/2	sin = 1 peak, slope cos = 0 (flat)	cos = 0, slope -sin = -1 (descending)

All match the curve shapes.

Two payoffs

• Small-angle: near 0, sin(x) ≈ x (slope at 0 is cos 0 = 1). First sliver of Taylor series.
• Oscillation: d²/dx²(sin x) = -sin x, so f'' = -f. This is the equation of springs, pendulums, sound, AC current, light. Why sine is everywhere in physics.

Why radians

The derivation needs unit speed, which needs arc length = angle, which is the definition of a radian. In degrees the point moves at speed π/180, and every trig derivative would carry an ugly π/180. Radians make calculus on trig clean.

Why it matters for AI

• Transformer positional encodings (Vaswani et al., 2017) use sin/cos waves of different frequencies to encode token position; differentiating through them uses these derivatives.
• 3D rotations (differentiable rendering, pose estimation) use sin/cos in rotation matrices.
• Signal processing / Fourier decomposes signals into sin/cos components.

Pitfalls to dodge

• Memorizing the pair (which gets the minus?). The picture decides: cosine’s coordinate shrinks as the point climbs, so the minus is on cosine.
• Forgetting radians. Clean derivatives hold only in radians.
• Treating trig as a separate topic. Same nudge-and-look method as the power rule, different shape.
• Confusing velocity with position. Position (cos x, sin x); velocity (the derivative) (-sin x, cos x).

The one-line version

Sine and cosine are a circling point’s coordinates, and their derivatives are its velocity, the position rotated a quarter turn, so sin -> cos and cos -> -sin.