Interactive lessons

Basic Trigonometry
from first principles

Each lesson is an interactive tool you can fiddle with — drag points, watch the geometry move, build intuition before touching a formula. This series grows from the Cartesian plane into the full unit-circle language that powers trigonometry.

Foundations

— 3 of 3 complete
(x, y) θ x y r Point Angle Triangle
Foundations — 01
Points and Angles on the Plane
Drag a point, connect it to the origin, measure the angle in standard position, and build the right triangle that introduces radius, x, and y.
interactive · 2D · draggable →
opp adj hyp sin = opp/hyp cos = adj/hyp tan = opp/adj
Foundations — 02
Ratios on the Triangle
Sine, cosine, and tangent as side ratios. Drag the point, watch the three ratios update live, and see why they depend only on the angle, not the size.
interactive · 2D · draggable →
sin θ cos θ r=1 (cos θ, sin θ) r = 1
Foundations — 03
The Unit Circle
Fix the radius at 1 and the coordinates of the point become cos θ and sin θ directly. Rotate the point through all four quadrants and watch the signs change.
interactive · 2D · draggable →

Coordinates & Identities

— 3 of 3 complete
r y x Cartesian (x,y) Polar (r, θ)
Coordinates — 04
Cartesian ↔ Polar
Map between (x, y) and (r, θ) interactively. Drag in either system and watch the other update — building the bridge that makes polar natural.
interactive · 2D · dual-mode →
(√3/2, 1/2) 1/2 √3/2 (√2/2, √2/2) (1/2, √3/2) 30° 45° 60°
Coordinates — 05
Special Angles
Derive the exact coordinates for 30°, 45°, and 60° from their triangles — no memorization, just geometry. Then extend them to all four quadrants.
interactive · 2D · exact values →
cos²θ sin²θ cos²θ + sin²θ = 1 Pythagoras
Identities — 06
The Pythagorean Identity
Why cos²θ + sin²θ = 1 is just Pythagoras applied to the unit circle. Drag the point, watch both squares resize, and see the sum stay constant.
interactive · 2D · proof-by-picture →

Graphs & Applications

— 3 of 3 complete
sin θ cos θ 2π ▶ animate
Graphs — 07
Sine & Cosine Graphs
Trace how a point moving around the unit circle unrolls into the sine and cosine waves. Period, amplitude, and phase shift become visually obvious.
interactive · 2D · animated →
−1 1 arcsin arccos + arctan
Graphs — 08
Inverse Trig Functions
Go backwards: given a ratio, find the angle. Explore arcsin, arccos, and arctan, and understand why restricted domains are necessary.
interactive · 2D · domain-restricted →
30° = 390° = 750°… + 2πk
Graphs — 09
Coterminal Angles & Periodicity
Why adding 360° (or 2π) returns you to the same point. Coterminal families, negative angles, and why trig functions repeat with period 2π.
interactive · 2D · animated →