Unit Circle Calculator (Free Exact Values for Any Angle)

A unit circle calculator finds the point where an angle’s terminal side meets the circle of radius 1, and that point is (cos θ, sin θ). For 150°, the point is (−√3/2, 1/2), so cos 150° = −√3/2 ≈ −0.8660 and sin 150° = 1/2. For 5π/4 (225°), it is (−√2/2, −√2/2), so tan(5π/4) = 1.

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What is the unit circle?

The unit circle is the circle with radius 1 centered at the origin, x² + y² = 1. Draw an angle θ in standard position: start on the positive x-axis and turn counterclockwise (clockwise for a negative angle). The terminal side crosses the circle at one point, and its coordinates are the cosine and sine of the angle: x = cos θ and y = sin θ. The other four functions follow from that point: tan θ = y/x, csc θ = 1/y, sec θ = 1/x and cot θ = x/y.

How to use the unit circle calculator

1. Angle → point: type any angle in degrees or radians (150, −45, 765, 5π/6 or pi/4), or drag the point around the circle. You get the exact point, the quadrant, the reference angle, coterminal angles and all six trig values.
2. Value → angles: choose sin θ, cos θ, tan θ or a point (x, y) and type a value such as 1/2 or −√3/2. You get every angle between 0° and 360° with its point, plus the general solution.
3. Unit circle chart: see all 16 special angles at once. Click any of them to open it in the calculator.

The unit circle chart

How to find the point for any angle

1. If the angle is negative or larger than 360°, add or subtract 360° until it is between 0° and 360°. Coterminal angles share the same point, so 400° ends at the same point as 40°.
2. Find the quadrant and the reference angle, the acute angle between the terminal side and the x-axis. The reference angle calculator explains this step in detail.
3. Take the first-quadrant point for the reference angle: (√3/2, 1/2) for 30°, (√2/2, √2/2) for 45° and (1/2, √3/2) for 60°.
4. Attach signs: x is negative in Quadrants II and III, and y is negative in Quadrants III and IV.

Angles that are not multiples of 30° or 45° have decimal coordinates. For 20°, the point is about (0.9397, 0.3420). Multiples of 15° still have exact values: sin 15° = (√6 − √2)/4.

Worked examples

150° and 60°

150° is in Quadrant II with reference angle 180° − 150° = 30°. The 30° point is (√3/2, 1/2); in Quadrant II x is negative, so the point is (−√3/2, 1/2). For 60° in Quadrant I the point is (1/2, √3/2), so cos 60° = 1/2, sin 60° = √3/2 ≈ 0.8660 and tan 60° = (√3/2)/(1/2) = √3.

5π/4 in radians

5π/4 × 180°/π = 225°, which is in Quadrant III with reference angle 45°. Both coordinates are negative: (−√2/2, −√2/2). So sin(5π/4) = cos(5π/4) = −√2/2 and tan(5π/4) = 1.

Negative and large angles: −45° and 405°

−45° + 360° = 315°, in Quadrant IV, so the point is (√2/2, −√2/2). 405° − 360° = 45°, so 405° ends at (√2/2, √2/2).

From a value back to angles: sin θ = −1/2

The y-coordinate is −1/2 at two points on the circle, in Quadrants III and IV, both with reference angle 30°: θ = 180° + 30° = 210° and θ = 360° − 30° = 330°. In radians these are 7π/6 and 11π/6. For more inverse problems, use the inverse trig calculator.

How to memorize the unit circle

Write the first-quadrant sines as √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. Simplified, that is 0, 1/2, √2/2, √3/2, 1. The cosines are the same list in reverse. Every other special angle reuses these numbers with signs from its quadrant, so you only need to memorize three points: (√3/2, 1/2), (√2/2, √2/2) and (1/2, √3/2).

For radians, the denominators tell you the family: angles over 6 (π/6, 5π/6, 7π/6, 11π/6) have reference angle 30°, over 4 have 45°, and over 3 have 60°.

Common unit circle mistakes

• Swapping x and y. The point is (cos θ, sin θ), cosine first. At 30° it is (√3/2, 1/2), not (1/2, √3/2).
• Wrong sign. Check the quadrant: cos 120° = −1/2 because x is negative in Quadrant II.
• Calculator in the wrong mode. sin 30 in radian mode gives about −0.9880, not 0.5.
• Only one solution. sin θ = 1/2 has two answers between 0° and 360°: 30° and 150°.

Unit circle practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you use the unit circle?

Find where the angle's terminal side meets the circle and read the coordinates: the x-coordinate is cos θ and the y-coordinate is sin θ. For 60°, the point is (1/2, √3/2), so cos 60° = 1/2, sin 60° = √3/2 and tan 60° = √3.

How do you calculate unit circle coordinates?

Use the reference angle and the quadrant. Take the first-quadrant point for the reference angle (30°, 45° or 60°) and give each coordinate the sign of the quadrant. For 150°, the reference angle is 30°, so the point is (−√3/2, 1/2).

How do you calculate unit circle values in radians?

Convert to degrees by multiplying by 180°/π, or use the denominator families: π/6-type angles use 30° values, π/4-type use 45° values and π/3-type use 60° values. For 5π/4 = 225°, the point is (−√2/2, −√2/2).

How do you find unit circle values on a calculator?

Set the calculator to degree or radian mode to match your angle, then press cos for the x-coordinate and sin for the y-coordinate. A calculator gives decimals, such as sin 60° ≈ 0.8660; this page also gives the exact value √3/2.

How do you get the unit circle on a TI-84 calculator?

Press MODE and choose PAR (parametric) and RADIAN. In Y=, enter X1T = cos(T) and Y1T = sin(T). In WINDOW set Tmin = 0, Tmax = 2π and Tstep = π/12, then press ZOOM and choose ZSquare. Press TRACE and move along the circle to read T, X = cos T and Y = sin T.

How do you solve the unit circle without a calculator?

Memorize the three first-quadrant points from the √n/2 pattern, find the reference angle, and attach the signs of the quadrant. That covers all 16 special angles, and the six functions follow from x and y.

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