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.
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)?
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?
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?
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)?
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.