Interactive Unit Circle Explorer

Drag point P(x,y) around the unit circle to explore trigonometric functions and their properties. This interactive app is a complement to a similar app in Unit Circle HTML5 Applet.

Go to Exploration Questions Go to Interactive Visualization Go to References
How to Explore
1
Drag point P around the unit circle
2
Observe how coordinates (x,y) relate to cos(θ) and sin(θ)
3
Watch tan(θ) = y/x and notice when it becomes undefined
4
Use Show All Graphs to compare functions simultaneously
5
Use Reset to return to initial position
Display Controls

Interactive Visualization

Unit Circle with Draggable Point P

Drag point P around the circle to change angle θ

Function Graphs: All Functions

sin(θ) = y-coordinate
cos(θ) = x-coordinate
tan(θ) = sin(θ)/cos(θ)
Current Values at Point P(θ)
Angle θ (anti-clockwise from positive x-axis)
0°
degrees
0.000
radians
cos(θ) = x-coordinate
1.000
sin(θ) = y-coordinate
0.000
tan(θ) = sin/cos
0.000
Point P Coordinates
(1.000, 0.000)
Asymptote Status
No asymptote at current position. Asymptotes occur at θ = π/2 (90°) and θ = 3π/2 (270°) where cos(θ) = 0, making tan(θ) = y/0 undefined.
Point P is at θ = 0° where cos(θ) = 1 ≠ 0

Exploration Questions & Answers

Focus on Understanding Domain, Range, and Asymptotes

Use the interactive visualization above to answer these questions about trigonometric functions.

  1. As you drag point P around the circle, can you find positions where point P doesn't exist? The x and y-coordinates are cos(θ) and sin(θ). What is the domain of sin(θ)? What is the domain of cos(θ)?
    View Answer
  2. Explore the x-intercepts, the maximums and minimums (if any) of the graphs of sin(θ) and cos(θ) by dragging point P to specific positions. Where do these occur?
    View Answer
  3. Drag point P around the circle. Can the x or y-coordinate ever be larger than 1 or smaller than -1? Why do sin(θ) and cos(θ) have a range between -1 and 1?
    View Answer
  4. Drag point P completely around the circle. After what angle do the values repeat? What is the period of sin(θ), cos(θ) and tan(θ)?
    View Answer
  5. tan(θ) is defined as the ratio y-coordinate / x-coordinate (sin/cos). Drag point P to positions where the x-coordinate equals zero. At what angles does this happen between 0 and 360°? What is the domain of tan(θ)?
    View Answer
  6. At the positions where tan(θ) is undefined, the graph shows asymptotic behavior. Explain why. What is the range of tan(θ)? Find all x-intercepts of tan(θ) between 0 and 360°. Explain their positions.
    View Answer
Key Trigonometric Concepts
Domain
Set of all possible input values (angles)
Range
Set of all possible output values
Asymptote
Line that the graph approaches but never touches
Periodicity
Function repeats its values at regular intervals

Additional Resources on Unit Circle and Trigonometric Functions