Maths / Trigonometry

Interactive Unit Circle: Exact Values and Practice Chart

Students and teachers can connect angles, triangles and trigonometric values with an interactive unit circle, then print or download a chart of all 16 special angles or a blank practice version.

Interactive Unit Circle: Exact Values and Practice Chart: Move the point or enter an angle to see how its right triangle relates to cosine, sine and tangent. Compare reference angles and quadrants, learn the exact fractions and square-root values for special angles, and print a labelled or blank chart for revision. Runs 100% locally in your browser with zero server file uploads.

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For students and teachers: drag the point to connect an angle with its triangle and trigonometric values, or enter any positive or negative angle.

xy0
Normalised angle30.0000°
Reference angle30.0000°
Quadrant1
Coordinates (cos, sin)(√3/2, 1/2)
tan θ√3/3
Radiansπ/6

Drag anywhere on the diagram, or focus the point and use arrow keys in one-degree steps. Exact values are shown at multiples of 30° and 45°; other angles show six decimal places. On an axis there is no quadrant, and tangent is undefined at 90° and 270°.

Unit circle · special angles0°0(1, 0)30°π/6(√3/2, 1/2)45°π/4(√2/2, √2/2)60°π/3(1/2, √3/2)90°π/2(0, 1)120°2π/3(−1/2, √3/2)135°3π/4(−√2/2, √2/2)150°5π/6(−√3/2, 1/2)180°π(−1, 0)210°7π/6(−√3/2, −1/2)225°5π/4(−√2/2, −√2/2)240°4π/3(−1/2, −√3/2)270°3π/2(0, −1)300°5π/3(1/2, −√3/2)315°7π/4(√2/2, −√2/2)330°11π/6(√3/2, −1/2)Coordinates are (cos θ, sin θ). A complete turn is 360° = 2π radians.

Unit circle and exact values

OpenStax, Precalculus 2e, Unit Circle: Sine and Cosine Functions: https://openstax.org/books/precalculus-2e/pages/5-2-unit-circle-sine-and-cosine-functions . A radius-one circle has coordinates (cos θ, sin θ). The 30°–60°–90° and 45°–45°–90° triangles provide the exact values shown.

Angle conventions

Start at the positive x axis and move anticlockwise to measure a positive angle. One complete turn is 360° or 2π radians. The reference angle tells you how far the point is from the x axis, from 0° to 90°. A point on an axis has no quadrant. Divide sine by cosine to find tangent; tangent is undefined when cosine is zero.

How to use it

  1. Enter an angle in degrees or radians, or drag the point around the circle.
  2. Read the reference angle, quadrant, coordinates and tangent.
  3. Choose the labelled or blank chart and print it or download SVG or PNG.

Privacy & limitations

Angle changes and chart exports run in your browser. Nothing is uploaded.

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Frequently asked questions

What happens on an axis?

An angle on the x or y axis has no quadrant. Tangent is undefined at 90° and 270° because cosine is zero. Reference angles on the x axis are 0° and on the y axis are 90°.

Can I enter negative angles or extra turns?

Yes. For example, −30° reaches the same point as 330°, and 765° reaches the same point as 45°. Switch between degrees and radians to express the same angle in either unit, including any extra turns.

What is on the printable chart?

All 16 distinct special angles in one turn, with degrees, fractions of π and exact cosine/sine coordinates. The blank version has spaces to practise these labels. PNG downloads have twice the preview resolution.

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