Calculus×Computing
ES

Trigonometric functions

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

sin⁡\sin and cos⁡\cos are the coordinates of a point turning around the unit circle. They describe every rotation and every oscillation, and therefore graphics, robotics, sound and signals.

Formulas

sin⁡2x+cos⁡2x=1,sin⁡′=cos⁡,cos⁡′=−sin⁡\sin^2 x + \cos^2 x = 1, \qquad \sin' = \cos, \quad \cos' = -\sin
R(θ)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}
rotation of the plane

Where it shows up in computing

  • Signal processing★★★★★fundamentalSignals, media and vision

    Pure tones are sinusoids; every signal is analysed as a sum of them.

  • Lighting and shading★★★★★frequentComputer graphics

    Lambert's law: diffuse light is proportional to cos⁡θ\cos\theta between normal and light direction.

  • Robot Jacobian (velocity kinematics)★★★★★frequentRobotics and control

    Forward kinematics of a revolute arm is sums of cos⁡\cos and sin⁡\sin of joint angles.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

ℒ AI and machine learning

3D Computer graphics

⚙ Robotics and control

⚛ Physics and simulation

∿ Signals, media and vision

⊕ Cryptography and security

ψ Quantum computing and physics

⇄ Optimization and systems

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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