liminfoUnit Circle Reference
Free web tool: Unit Circle Reference
Unit Circle Reference
Interactive unit circle with trig values for standard angles. Click any point.
| Deg | Rad | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undef |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -1/√3 |
| 180° | π | 0 | -1 | 0 |
| 210° | 7π/6 | -1/2 | -√3/2 | 1/√3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 |
| 270° | 3π/2 | -1 | 0 | undef |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -1/√3 |
| 360° | 2π | 0 | 1 | 0 |
About Unit Circle Reference
The Unit Circle Reference is an interactive SVG-based visualization of the mathematical unit circle, displaying exact trigonometric values for all 17 standard angles from 0° to 360°. The supported angles are the commonly tested values in trigonometry: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, and 360°. For each angle, the tool shows the degree measure, the equivalent radian expression (in terms of π fractions like π/6, π/4, π/3, etc.), and the exact values of sin(θ), cos(θ), and tan(θ) as simplified radical fractions rather than decimal approximations.
This tool is primarily used by trigonometry and precalculus students who need to memorize or verify unit circle values for quizzes and exams. It is also useful for calculus students who need quick access to trig values when computing derivatives, integrals, or evaluating trigonometric identities. The interactive nature — click any dot on the circle to highlight it and display its values — makes it significantly more engaging than a static table, helping with active recall and memorization. The accompanying reference table below the circle allows users to scan all values at once.
The tool renders a 360×360 SVG with the unit circle drawn as a circle of radius 140px centered at (180, 180). Axis lines run horizontally and vertically through the center. Each standard angle is calculated using Math.cos and Math.sin to position a clickable dot on the circle. When a dot is selected, a line is drawn from the center to that point in blue, the dot enlarges from radius 4 to 6, and a 2×2 grid of value cards (angle, sin, cos, tan) appears to the right. The full reference table below uses click events on rows to sync with the circle visualization.
Key Features
- 17 standard angles from 0° to 360° plotted on an interactive SVG unit circle
- Click any angle point to highlight it, draw the radius line, and display all trig values
- Exact values shown as simplified radicals (e.g., √3/2, √2/2, 1/√3) not decimals
- Radian equivalents displayed as π fractions (π/6, π/4, π/3, π/2, 2π/3, etc.)
- Synchronized reference table — clicking a row highlights the corresponding circle point
- Four-card display for selected angle: degree + radian, sin(θ), cos(θ), tan(θ)
- tan(θ) correctly shows "undef" for 90° and 270° where tangent is undefined
- Dark mode support and responsive layout with side-by-side circle and table on large screens
Frequently Asked Questions
What angles are included in this unit circle reference?
The reference includes all 17 standard angles used in trigonometry: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, and 360°. These correspond to the angles in the first through fourth quadrants that appear on most trig exams and problem sets.
Why are the trig values shown as fractions instead of decimals?
Exact fractional values like √3/2 and √2/2 are the standard form used in mathematics and physics. They are exact and do not introduce rounding errors. These are the values students are expected to memorize and recognize on exams, and they are directly usable in algebraic manipulations.
How do I use the interactive unit circle?
Click any gray dot on the circle SVG to select that angle. A blue line will be drawn from the center to the point, the dot will enlarge, and four value cards (showing the angle in degrees and radians, plus sin, cos, and tan values) will appear next to the circle. You can also click any row in the reference table below to select the same angle.
Why is tan(90°) and tan(270°) shown as undefined?
Tangent is defined as sin(θ)/cos(θ). At 90° and 270°, cos(θ) equals 0, which would result in division by zero. Therefore, the tangent function is undefined at these angles. On a graph, the tangent function has vertical asymptotes at these points.
What is the relationship between degrees and radians?
Radians and degrees are two units for measuring angles. One full rotation is 360° = 2π radians. To convert degrees to radians, multiply by π/180. For example, 30° = 30 × (π/180) = π/6. The unit circle reference shows both representations for each standard angle.
How do I remember the sin and cos values for the standard angles?
A common memory trick: for the first quadrant angles (0°, 30°, 45°, 60°, 90°), the sin values are √0/2, √1/2, √2/2, √3/2, √4/2 (simplified: 0, 1/2, √2/2, √3/2, 1). The cos values are the reverse of this sequence. For other quadrants, use the reference angle and apply the sign based on which quadrant you are in.
What are the signs of sin, cos, and tan in each quadrant?
In quadrant I (0°–90°): all positive. In quadrant II (90°–180°): sin is positive, cos and tan are negative. In quadrant III (180°–270°): tan is positive, sin and cos are negative. In quadrant IV (270°–360°): cos is positive, sin and tan are negative. A common mnemonic is "All Students Take Calculus" (ASTC) for Quadrants I, II, III, IV.
Can I use this as a study aid to memorize the unit circle?
Yes, this is an excellent memorization aid. You can quiz yourself by clicking each angle on the circle and checking if you can recall the sin, cos, and tan values before they are revealed in the cards. The table also lets you scan all values at once and identify patterns — for instance, sin(30°) = cos(60°) and sin(45°) = cos(45°) = √2/2.