Math
Unit Circle Calculator
The unit circle is the circle of radius 1 centered at the origin.
Any angle θ drawn from the positive x-axis meets it at the point (cos θ, sin θ). Choose one of the special angles or type your own to see the exact coordinates, the radian measure as a π fraction, all six trig functions, and the full reference table.
Every multiple of 15° plus the 18° family (18°, 36°, 54°, 72°, ...).
Try an example
Result
Point (cos θ, sin θ)
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- Radians
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- sin θ
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- cos θ
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- tan θ
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- Quadrant
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- Angles with this value
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the unit circle tells you
Because the radius is 1, the coordinates of the terminal point are the cosine and sine of the angle. That single fact gives you every trig function: tan θ is y / x, and the The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. functions flip those fractions.
How to read the unit circle
- Angles are measured counterclockwise from the positive x-axis. Negative angles go clockwise.
- One of the four regions the x- and y-axes cut the coordinate plane into. I points have both coordinates positive; Quadrant II has negative x; Quadrant III has both negative; Quadrant IV has negative y.
- Angles with the same The small positive angle between a given angle's arm and the x-axis. share the same coordinates up to sign: 30°, 150°, 210°, and 330° all use 1/2 and √3/2.
- The 18° family (18°, 36°, 54°, 72°) comes from the regular pentagon and involves √5.
Degrees and radians on the circle
A full circle is 360° or 2π An angle unit where a full turn is 2π instead of 360 degrees., so 180° = π, 90° = π/2, 60° = π/3, 45° = π/4, and 30° = π/6. Multiply degrees by π/180 to get radians.
Going backward: from a value to the angle
The unit circle also answers questions like 'which angles have sin θ = 1/2?'. Find the angle whose coordinate matches in Quadrant I, then use symmetry: sine repeats at 180° - θ, cosine at 360° - θ, and tangent every 180°. So sin θ = 1/2 at 30° and 150°, cos θ = 1/2 at 60° and 300°, and tan θ = 1 at 45° and 225°. Choose Find angles from a trig value to see both solutions on the circle and the general solution with k full turns.
A hand trick for the first quadrant
Hold up a hand and number the fingers 0, 1, 2, 3, 4 from thumb to pinky for 0°, 30°, 45°, 60°, 90°. For sin θ take √(finger number)/2; for cos θ count from the other end. That gives sin 30° = √1/2 = 1/2, sin 45° = √2/2, sin 60° = √3/2, and sin 90° = √4/2 = 1. Reference angles and quadrant signs then give every other special angle.
How to use this calculator
- Pick a special angle from the list to see its exact coordinates, or switch to Type any angle for decimals, negatives, or radians.
- To go the other way, choose Find angles from a trig value, pick the function, and type the value (1/2, -√3/2, 0.7071). Both solutions in one turn appear on the circle.
- Read the point (cos θ, sin θ) in the header and the six functions in the grid.
- Find your angle highlighted in the special-angle table to compare it with its neighbors.
Common mistakes and edge cases
- Swapping the coordinates: the point is (cos θ, sin θ), with cosine first.
- Forgetting that tan is undefined where x = 0 (90° and 270°) and cot is undefined where y = 0 (0° and 180°).
- Reading the table for 210° and dropping the negative signs: Quadrant III points have both coordinates negative.
- Typing a rounded radian value: 0.7854 is not exactly π/4, so the result is a decimal. Type π/4 for exact coordinates.
Worked examples
150° from the special list
Point (cos θ, sin θ)
(-√3/2, 1/2) ≈ (-0.866025, 0.5)
Quadrant III angle
225° has both coordinates negative
Point (cos θ, sin θ)
(-√2/2, -√2/2) ≈ (-0.707107, -0.707107)
Custom radians
θ = 7π/6 typed as radians
Point (cos θ, sin θ)
(-√3/2, -1/2) ≈ (-0.866025, -0.5)
Negative custom angle
θ = -90° lands at (0, -1)
Point (cos θ, sin θ)
(0, -1) ≈ (0, -1)
Non-special angle
θ = 20° gives decimal coordinates
Point (cos θ, sin θ)
(0.939693, 0.34202)
18° family
θ = 72°
Point (cos θ, sin θ)
((√5 − 1)/4, √(10 + 2√5)/4) ≈ (0.309017, 0.951057)
Which angles have sin θ = 1/2?
30° and 150°
Point (cos θ, sin θ)
(√3/2, 1/2) ≈ (0.866025, 0.5)
Which angles have cos θ = -√2/2?
135° and 225°
Point (cos θ, sin θ)
(-√2/2, √2/2) ≈ (-0.707107, 0.707107)
Which angles have tan θ = √3?
60° and 240°
Point (cos θ, sin θ)
(1/2, √3/2) ≈ (0.5, 0.866025)
Impossible value
sin θ = 2 has no solution
Point (cos θ, sin θ)
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Bad input
Point (cos θ, sin θ)
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Frequently asked questions
What is the unit circle used for?+
It defines sine and cosine for every angle, not only the acute angles of a right triangle. The x-coordinate is cosine and the y-coordinate is sine, which makes the signs, symmetries, and periodic behavior of the trig functions visible.
How do I memorize the unit circle?+
Learn the first quadrant: 30° gives (√3/2, 1/2), 45° gives (√2/2, √2/2), 60° gives (1/2, √3/2). Every other special angle reuses those numbers with quadrant signs, and the reference angle tells you which pair to use.
Why are the coordinates (cos θ, sin θ) and not (sin θ, cos θ)?+
Cosine is the horizontal (x) distance and sine is the vertical (y) distance from the origin. Coordinates are written (x, y), so cosine comes first.
What are the coordinates at 90°?+
(0, 1). Cosine is 0 and sine is 1, so tan 90° = 1/0 is undefined.
Where do the 18° values come from?+
From the regular pentagon and the golden ratio. cos 36° = (1 + √5)/4 and sin 18° = (√5 − 1)/4, and the rest of the family follows from cofunction and reference-angle rules.
Can I use radians?+
Yes. Switch to Type any angle, choose Radians, and enter π/6, 5pi/6, 2π/3, or a decimal. The special-angle list also shows each angle's radian form.
What is tan 30° on the unit circle?+
tan θ = y / x = sin θ / cos θ. At 30° the point is (√3/2, 1/2), so tan 30° = (1/2) / (√3/2) = 1/√3 = √3/3 ≈ 0.5774.
How do I find cosecant, secant, and cotangent from the unit circle?+
Flip the coordinates: csc θ = 1/y, sec θ = 1/x, cot θ = x/y. At 150° the point is (-√3/2, 1/2), so csc 150° = 2, sec 150° = -2/√3 = -2√3/3, and cot 150° = -√3. Whenever a coordinate is 0 the matching reciprocal is undefined.
How do I find arcsin(1/2) with the unit circle?+
Look for the angles whose y-coordinate is 1/2: 30° and 150°. arcsin returns the one in [-90°, 90°], so arcsin(1/2) = 30° = π/6, and 150° is the other solution of sin θ = 1/2 in one full turn. The Find angles from a trig value mode shows both.
Why does every angle have two solutions in one turn?+
Because each coordinate is shared by two points on the circle: two points have the same height (same sine) and two have the same x-position (same cosine). Only the extremes ±1 belong to a single point, which is why sin θ = 1 has just θ = 90°.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026