Lesson 2 of 9 · Trigonometry foundation

An angle is a distance around a circle.

Your win: turn any angle into the coordinates \((\cos\theta,\sin\theta)\) and recognize one full cycle as \(2\pi\) radians.

12 minutesNeeds: coordinatesOutcome: read sine and cosine geometrically

Mission link: RoPE gives pair \(i\) at token position \(m\) the angle \(m\theta_i\). Trigonometric functions convert that angle into usable coordinates.

Why radians?

Radians measure an angle by arc length ÷ radius. On a circle of radius 1, the angle simply equals the distance traveled around its edge. The circumference is \(2\pi\), so one complete turn is \(2\pi\) radians.

Turn	Radians	Degrees
Quarter	\(\pi/2\)	\(90^\circ\)
Half	\(\pi\)	\(180^\circ\)
Full	\(2\pi\)	\(360^\circ\)

Cosine and sine are coordinates

Begin at \((1,0)\) on the unit circle. Turn counterclockwise by \(\theta\). The endpoint is:

\[(x,y)=(\cos\theta,\sin\theta)\]

This is the entire trigonometric picture RoPE needs. Cosine is the horizontal coordinate; sine is the vertical coordinate.

Rotate from \((1,0)\) 90° · 1.57 rad

The identity that protects length

Every endpoint on the unit circle is exactly one unit from the origin. Substitute \((\cos\theta,\sin\theta)\) into the distance formula:

\[\cos^2\theta+\sin^2\theta=1\]the Pythagorean identity

This identity is the engine behind length-preserving rotation matrices.

Predict before checking the diagram

What point corresponds to \(\theta=\pi\)?

Practice before moving on

Convert a quarter, half, and three-quarter turn to radians.
Give \((\cos\theta,\sin\theta)\) for \(\theta=0,\pi/2,\pi,3\pi/2\).
If \(\cos\theta=3/5\) in quadrant I, calculate \(\sin\theta\).
A pair advances \(0.2\) radians per token. What angle has it reached at position \(m=7\)?
Estimate its period in tokens using \(2\pi/\theta\).

Check solutions

\(\pi/2,\pi,3\pi/2\).
\((1,0),(0,1),(-1,0),(0,-1)\).
\(\sin\theta=\sqrt{1-9/25}=4/5\).
\(7(0.2)=1.4\) radians.
\(2\pi/0.2=10\pi\approx31.4\) tokens.

Token-period connection: if a pair advances by \(\theta\) radians per token, it completes a full turn after approximately \(2\pi/\theta\) tokens.

Primary source: OpenStax 5.2: Unit Circle, Sine, and Cosine. Focus on the definitions and the Pythagorean identity.

I can read \((\cos\theta,\sin\theta)\) as a location on the unit circle.

Ask the teaching agent to draw a particular angle or explain radians using a different analogy if the arc-length definition is not yet intuitive.