6. Trigonometry

Trigonometric Unit Circle Visualizer

Rotate a vector ray on a unit circle (r = 1). Observe circular coordinates (cos θ, sin θ) and watch waveforms plot dynamically in real time.

Virtual Viewport
sin(θ) [y]
0.866
cos(θ) [x]
0.500
tan(θ) [y/x]
1.732

Active Sliders

θAngle of Rotation60°
0360

Angle in degrees relative to the positive x-axis.

Syllabus Objectives

  • Explain sine, cosine, and tangent coordinates on a unit circle
  • Identify signs of trigonometric ratios in different quadrants (CAST system)
  • Relate unit circle rotation to periodic sine and cosine wave graphs.

Mathematical Equations

Unit Circle Radius
x² + y² = 1
Trigonometric Ratios
sin(θ) = y,   cos(θ) = x,   \tan(θ) = yx

Inquiry Guide Sheet

1

Set angle to 30°. Observe the y-coordinate value (0.5). What trigonometric ratio is this? (sin 30°)

2

Rotate to 150°. Observe why sin is still positive but cos becomes negative.

3

Vary angle to 90° and 270°. Identify what happens to tangent (vertical line goes to infinity).