Math Visualizations

What is the Unit Circle?

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. It provides a powerful visual framework for understanding trigonometric functions.

Any point on the unit circle can be described by an angle $\theta$ (measured from the positive x-axis). The coordinates of that point are:

$(\cos\theta, \sin\theta)$

This means the x-coordinate gives $\cos\theta$ and the y-coordinate gives $\sin\theta$ directly.

The Six Trigonometric Functions

From the unit circle, all six trigonometric functions can be defined:

• $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = y$
• $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = x$
• $\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}$
• $\csc\theta = \frac{1}{\sin\theta}$
• $\sec\theta = \frac{1}{\cos\theta}$
• $\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{x}{y}$

Geometric Constructions

Toggle the tan/sec and csc/cot layers to see geometric constructions on the circle:

• tan θ is the length of the tangent segment from (1,0) to where the extended radius meets the vertical tangent line at x=1
• sec θ is the distance from the origin to that intersection point
• csc θ is the distance from (0,1) to where the extended radius meets the horizontal tangent line at y=1
• cot θ is the distance from the origin to that intersection point

Wave Plots

The wave plot shows how each function varies as θ goes from 0° to 360°. Toggle individual functions to see:

• Sine and cosine oscillate between -1 and 1
• Tangent and secant have vertical asymptotes at 90° and 270° (where cos θ = 0)
• Cotangent and cosecant have vertical asymptotes at 0°, 180°, and 360° (where sin θ = 0)

Try This

1. Drag the point around the circle and watch how the values change
2. Toggle tan/sec and observe where the construction extends beyond the circle
3. Enable wave curves for tan, cot, sec, csc and note the asymptote markers
4. Click angle ticks for special angles like 30°, 45°, 60°, 90° to see exact values