WorkedMathGlossary

Unit Circle Calculator

Enter any angle in degrees or radians to get its cosine, sine and tangent, plus the quadrant, reference angle, and the full 16-angle reference table.

cos
0.7071
√2/2
sin
0.7071
√2/2
tan
1.0000
1

Angle (0-360°): 45.00° = 0.7854 rad (exactly π/4)

Quadrant: 1, reference angle 45°

The 16 standard unit circle angles

DegreesRadianscossintan
0°0100
30°π/6√3/21/2√3/3
45°π/4√2/2√2/21
60°π/31/2√3/2√3
90°π/201undefined
120°2π/3−1/2√3/2−√3
135°3π/4−√2/2√2/2−1
150°5π/6−√3/21/2−√3/3
180°π−100
210°7π/6−√3/2−1/2√3/3
225°5π/4−√2/2−√2/21
240°4π/3−1/2−√3/2√3
270°3π/20−1undefined
300°5π/31/2−√3/2−√3
315°7π/4√2/2−√2/2−1
330°11π/6√3/2−1/2−√3/3

How the Unit Circle Works

Start at (1, 0) on the positive x-axis and sweep counterclockwise by angle θ. The point you land on has x-coordinate cos θ and y-coordinate sin θ — always, because the circle has radius 1. Tangent is just the ratio sin θ / cos θ, which is also the slope of the line from the origin to that point.

Radians measure the same angle as an arc length: one full trip around the unit circle is 2π radians (360°), so π radians is a half-turn (180°) and π/2 radians is a quarter-turn (90°). Converting between the two is just θ_rad = θ_deg × (π / 180).

Worked example

Find cos, sin and tan of 210°.

210° is in quadrant 3 (between 180° and 270°), where sin and cos are both negative but tan is positive.

Its reference angle is 210° − 180° = 30°, and 30°'s values are cos = √3/2, sin = 1/2.

Applying quadrant 3's signs: cos 210° = −√3/2, sin 210° = −1/2, tan 210° = sin/cos = √3/3

Common mistake

Mixing up which coordinate is which: it's cos θ = x and sin θ = y, not the reverse. A quick check: at 0°, you're at (1, 0) — cos 0° = 1 (the x-value) and sin 0° = 0 (the y-value). At 90°, you're at (0, 1) — cos 90° = 0 and sin 90° = 1, the exact opposite pairing from 0°.

Keep going

  • The unit circle is a circle with radius fixed at 1; the circle calculator works with any radius to find circumference, area, and arc length. Circle Calculator
  • An angle on the unit circle also marks off an arc; the arc length calculator finds that arc's real length once the circle's radius isn't fixed at 1. Arc Length Calculator

Frequently Asked Questions

What is the unit circle?

A circle with radius 1 centered at the origin. For any angle measured counterclockwise from the positive x-axis, the point where that angle crosses the circle has coordinates (cos θ, sin θ) — which is exactly why cosine and sine of an angle are defined the way they are.

How many angles are on a standard unit circle?

16 distinct angles between 0° and 360° (not counting 360° itself, which repeats 0°): every multiple of 30° (0, 30, 60, 90, 120...) and every multiple of 45° (0, 45, 90, 135...), for 12 + 8 minus the 4 they share = 16 total. Those are the angles with clean, exact cos/sin/tan values worth memorizing.

What is a reference angle and why does it matter?

A reference angle is the acute angle (always between 0° and 90°) between the terminal side of your angle and the x-axis. Every angle's cos, sin and tan have the same absolute value as its reference angle's — only the sign changes depending on the quadrant — so memorizing the first-quadrant values (0°, 30°, 45°, 60°, 90°) and the quadrant sign rules covers all 16 standard angles.

How do I remember the sign of sin, cos and tan in each quadrant?

"All Students Take Calculus," reading quadrants 1 through 4: quadrant 1 has All three positive, quadrant 2 only Sine positive, quadrant 3 only Tangent positive, quadrant 4 only Cosine positive. Quadrant 2 example: 150° has sin = 1/2 (positive) but cos = −√3/2 and tan = −√3/3 (both negative).

Why is tan undefined at 90° and 270°?

Tangent is sin/cos, and cos = 0 at exactly those two angles, making the division undefined the same way any division by zero is. Approaching 90° from either side, tan grows without bound — from below (89.9°) it shoots toward positive infinity, from above (90.1°) toward negative infinity.

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