Unit Circle Chart
A clickable unit circle with every standard angle marked in degrees and radians and the exact (cos θ, sin θ) written along its radius. Switch between the usual 30° and 45° angles and every 15°, add tangent values and quadrant sign notes, highlight an angle to see all six trig values, then print the chart or save it as an SVG or PNG. Free, no sign-up.
The chart prints on one page and the controls stay off the paper. Tick the table option for a second page of values.
Values at the selected angle
Select a point on the chart, or choose an angle above, to see all six trig values.
How to Use the Unit Circle Chart
- 1. Choose the angles. Multiples of 30° and 45° is the standard sixteen-point chart; Every 15° adds the eight angles between them, whose values involve (√6 − √2)/4.
- 2. Choose what to show. Tick degrees, radians, the coordinates (cos θ, sin θ), tangent values and the quadrant sign notes in any combination. A cleaner chart is easier to fill in as a worksheet.
- 3. Click an angle. Pick any point on the circle, or choose it from the list, to draw its right triangle and see sine, cosine, tangent, cosecant, secant and cotangent, exact and as decimals, with its quadrant and reference angle.
- 4. Print or save. Print puts the chart alone on one page; add the table for a second page of values. The SVG keeps its sharpness at any size, and the PNG is ready for a document.
What the Unit Circle Is
The unit circle is the circle of radius 1 centered at the origin, with equation x² + y² = 1. Walk an angle θ counterclockwise from the positive x-axis and you land on the point (cos θ, sin θ): the x-coordinate is the cosine and the y-coordinate is the sine. The tangent is the ratio y / x, which is why it is undefined wherever x = 0, at 90° and 270°. Because every point is on the circle, cos² θ + sin² θ = 1 — the Pythagorean theorem in the right triangle the chart draws.
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
There is a pattern to remember: the sines of 0°, 30°, 45°, 60° and 90° are √0/2, √1/2, √2/2, √3/2 and √4/2, and the cosines are the same list backward. Every other angle on the chart reuses these five values.
From the First Quadrant to the Whole Circle
Every special angle has a reference angle in the first quadrant: the acute angle its terminal side makes with the x-axis (150° has reference angle 30°, 225° has 45°, 330° has 30°). The values match the reference angle's, up to sign, and the quadrant decides the signs: all three are positive in quadrant I, sine in II, tangent in III and cosine in IV. The note "All Students Take Calculus" lists the positive functions in order.
- sin 150°. The reference angle is 30° and sine is positive in quadrant II, so sin 150° = 1/2.
- cos 5π/4. 5π/4 is 225°, in quadrant III with reference angle 45°. Cosine is negative there: cos 5π/4 = −√2/2.
- tan 120°. 120° is in quadrant II with reference angle 60°. Tangent is negative: tan 120° = −√3.
- Degrees to radians. Multiply by π/180: 210° × π/180 = 7π/6.
Printing the Chart
The chart is drawn on a white background at a fixed proportion and sized to fill one page, so it prints the same on Letter and A4. Use the browser's print dialog, or choose Save as PDF for a file. Turn off the coordinates and tangent values for a blank chart that students fill in, or leave them on for a reference sheet.
Frequently Asked Questions
- What is the unit circle?
- The unit circle is the circle with radius 1 centered at the origin of the coordinate plane. A point on it at angle θ from the positive x-axis has coordinates (cos θ, sin θ), which lets the circle define sine and cosine for every angle, not only the acute angles of a right triangle.
- How do you memorize the unit circle?
- Learn the first quadrant: the angles 30°, 45° and 60° have sines 1/2, √2/2 and √3/2 — the square roots of 1, 2 and 3, over 2 — and the cosines run the other way. Then find every other angle from its reference angle and the quadrant signs, instead of memorizing sixteen points.
- What are the coordinates at 30°, 45° and 60°?
- At 30° the point is (√3/2, 1/2); at 45° it is (√2/2, √2/2); at 60° it is (1/2, √3/2). Each is on the circle: for 30°, (√3/2)² + (1/2)² = 3/4 + 1/4 = 1.
- How do you convert degrees to radians?
- Multiply by π/180, since 180° is π radians. So 90° is π/2, 60° is π/3 and 270° is 3π/2. To go back, multiply radians by 180/π.
- Why is tan 90° undefined?
- Because tan θ = sin θ / cos θ and cos 90° = 0, so the quotient divides by zero. Geometrically, the tangent is read off the vertical line x = 1; the terminal side at 90° is itself vertical, so it runs parallel to that line and never meets it. The same happens at 270°.
- How do you find sine and cosine for angles above 90°?
- Find the reference angle — the acute angle to the x-axis — look up its value, and attach the sign for the quadrant. For 210° the reference angle is 30° and both sine and cosine are negative, so sin 210° = −1/2 and cos 210° = −√3/2.
- Why does the chart have an every-15° option?
- Angles such as 15° and 75° also have exact values, built from √6 and √2: sin 15° = (√6 − √2)/4 and cos 15° = (√6 + √2)/4. They appear on some exams and in sum-and-difference formula problems, so the chart can include them.