Module 2: Trigonometry & Angles

Welcome to Module 2: Trigonometry & Angles. Module 1 built everything reachable without angles. This module supplies the missing tool and immediately spends it: right-triangle ratios and the unit circle, then rotation matrices, then the coordinate-frame transforms that field-oriented drive and every vision pipeline depend on, and finally 4-quadrant headings, circle topology and 3D quaternions.

Concepts in this Module

• Concept 01: The Unit Circle & Trigonometric Ratios
  • The Everyday Problem: A wheel is pointed 30° off down-field, moving at 4.0 m/s. How much of that carries the robot down-field, and how much sideways?
  • Code & Math: Why the ratios depend on the angle alone, SOH-CAH-TOA, the hypotenuse-1 collapse onto the unit circle, signs across all four quadrants, radians as arc length, and sin² + cos² = 1 as Pythagoras.
• Concept 02: Rotating a Vector
  • The Everyday Problem: The driver pushes “away from me” but the robot is turned 40 degrees. How does that command become wheel speeds?
  • Code & Math: Rotation derived by tracking where î and ĵ land, proof that rotation preserves length, undoing a rotation by flipping one sign, and composition deriving the angle-addition identities.
• Concept 03: Coordinate Frames (Field, Robot & Camera)
  • The Everyday Problem: The camera sees a game piece 1.5 meters ahead. Where is it on the field?
  • Code & Math: Rigid transforms combining rotation and translation, chaining field to robot to camera, and inverting a transform.
• Concept 04: Inverse Trig & 4-Quadrant Heading with atan2
  • The Everyday Problem: You know the two components. How do you get the angle back — without confusing North-East for South-West?
  • Code & Math: Why each inverse trig function needs a restricted range, how tan⁻¹(y/x) collapses two quadrants into one, and how atan2(y, x) recovers the sign information that the division destroyed.
• Concept 05: Angle Wrapping & Shortest Angular Distance
  • The Everyday Problem: A wheel at 350° is told to go to 10°. Why does it spin 340° the wrong way for a 20° move?
  • Code & Math: Angles live on a circle rather than a line, folding a difference into ±180°, why interpolating angles needs the wrap first, and the 90° threshold behind the swerve 180° flip.
• Concept 06: Law of Sines, Law of Cosines & Two-Link Arms
  • The Everyday Problem: A jointed arm forms a triangle with no right angle in it. Everything so far assumed one.
  • Code & Math: The law of cosines derived as a generalization of Pythagoras, solving a triangle from three sides, two-link inverse kinematics, reachability limits, and the elbow-up/elbow-down ambiguity.
• Concept 07: 3D Rotations, Gimbal Lock & Quaternions
  • The Everyday Problem: Yaw alone describes a robot on flat carpet. A robot on a ramp, or an arm on two axes, needs more.
  • Code & Math: Why 3D rotations depend on order, gimbal lock as a lost degree of freedom, Euler’s rotation theorem, and quaternions as an axis-angle encoding.