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Unit Circle Calculator

Drag the angle, see exact trig values — AI explains everything

This interactive unit circle calculator lets you explore every angle from 0° to 360°. Drag the θ slider and watch the blue point trace the unit circle x^2 + y^2 = 1 on a polar grid.

The trig readout panel (top-left) shows exact values: sin θ, cos θ, tan θ, the quadrant, and the radian equivalent. At key angles (30°, 45°, 60°, 90°, ...) you'll see the famous exact fractions like √2/2 and √3/2.

The muted dots mark the key angles for reference. Ask the AI assistant anything: "What's sin(150°)?", "Why is tan(90°) undefined?", "Show me the reference triangle at 225°." The AI draws directly on the graph to explain.

Graph

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FAQ

What is the unit circle?
The unit circle is a circle centered at the origin (0, 0) with a radius of exactly 1. Its equation is x^2 + y^2 = 1. Every point on it can be written as (cos θ, sin θ).
What are the exact values at key angles?
At 30°: (√3/2, 1/2). At 45°: (√2/2, √2/2). At 60°: (1/2, √3/2). At 90°: (0, 1). These values repeat with sign changes in other quadrants. Memorize the first quadrant and you know them all.
Why is tangent undefined at 90° and 270°?
Tangent = sin θ / cos θ. At 90° and 270°, cos θ = 0, so you're dividing by zero. Geometrically, the tangent line at the top and bottom of the circle is vertical — it has no finite slope.
What is a reference angle?
The reference angle is the acute angle (0°–90°) between the terminal side of your angle and the x-axis. For example, 150° has reference angle 30°, and 225° has reference angle 45°. Trig values of the original angle equal those of the reference angle, with signs determined by the quadrant.
How do I convert between degrees and radians?
Multiply degrees by \pi/180 to get radians. So 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6. A full circle is 360° = 2π radians.

Related Topics

Unit Circle LessonSine & CosinePythagorean Theorem