Mathematics calculator

Unit Circle Calculator

Find the point (cos θ, sin θ), tangent value, and quadrant for an angle measured in degrees.

Calculator inputs

Enter your values

Unit-circle coordinates
Calculated output

Result summary

Your result will appear here

Complete the required fields and select the calculate button.

Unit circle with x and y axes, radius, angle theta, and coordinate point
On a unit circle, the point at angle θ has coordinates x = cos θ and y = sin θ.

Table of Contents

1

What is Unit Circle?

The unit circle calculator takes an angle in degrees and returns the corresponding x-coordinate, y-coordinate and tangent. On a circle of radius 1 centred at the origin, the point is (cos θ, sin θ). Tangent is sin θ divided by cos θ when cosine is not zero.

Angles may be positive, negative or larger than one full turn. Trigonometric values repeat every 360°, so coterminal angles land on the same point. The page rounds finite values to six decimal places and reports tangent as undefined on vertical terminal lines.

2

How to Use This Calculator

  1. Enter an angle measured in degrees.
  2. Select Calculate.
  3. Read x = cosine and y = sine.
  4. Read tangent if it is defined.
  5. Use exact special-angle values when symbolic work requires them; the displayed decimals are rounded.

Positive angles rotate counterclockwise from the positive x-axis; negative angles rotate clockwise. The calculation converts degrees to radians internally because standard trigonometric functions use radians. The coordinate display was corrected so cosine appears with x and sine with y.

3

Formula Used

x = cos θ, y = sin θ, and tan θ = sin θ ÷ cos θ when cos θ ≠ 0. Every point satisfies x² + y² = 1.

Degrees convert by radians = degrees × π/180. Sine and cosine repeat every 360°. Tangent repeats every 180° and is undefined at 90° + 180°k for any integer k.

4

Variables Explained

Symbol or termMeaning
θEntered angle in degrees.
xHorizontal coordinate, equal to cos θ.
yVertical coordinate, equal to sin θ.
tan θSlope y/x where x is nonzero.
Unit circleCircle of radius 1 centred at (0,0).
Coterminal anglesAngles differing by whole multiples of 360°.
5

Worked Example

Enter 150°. Its reference angle is 30° and the point lies in quadrant II, where cosine is negative and sine is positive. Thus cos 150° = −√3/2 ≈ −0.866025 and sin 150° = 1/2 = 0.5.

Tangent is 0.5 ÷ −0.866025… ≈ −0.57735. The endpoint reports x = −0.866025, y = 0.5, tan = −0.57735. As a check, x² + y² is approximately 1, allowing for rounding.

6

Why Use This Calculator?

The signs identify the quadrant: (+,+) in I, (−,+) in II, (−,−) in III and (+,−) in IV. Axis angles have a zero coordinate and do not belong to a quadrant. At 90° and 270°, x is zero, so tangent is undefined rather than an extremely large finite number.

Rounded decimals are convenient for numerical work but can obscure exact patterns. For geometry and algebra, √2/2 or √3/2 may communicate more than 0.707107 or 0.866025. Floating-point results near zero are rounded for display; use identities and exact values when proving a result.

The calculator supplies trigonometric values only. It does not solve triangles where side lengths, inverse functions or ambiguous cases require additional information.

Reference angles help verify results without memorising every coordinate. Reduce the angle to a position from 0° through 360°, identify the acute angle to the x-axis, find its familiar sine and cosine magnitudes, then assign signs from the quadrant. At 150°, the reference angle is 30°, which explains the 1/2 and √3/2 magnitudes.

Coordinates also connect the unit circle to periodic motion. The x-coordinate can model horizontal projection and y the vertical projection as a point rotates. Scaling a circle to radius r multiplies both coordinates by r, but the sine and cosine values themselves remain dimensionless ratios. Angle units must still be stated clearly.

The Pythagorean identity is a strong numerical check: cos²θ + sin²θ should equal 1, apart from rounding. Tangent has no such bounded range; it can grow rapidly as cosine approaches zero from either side. That behaviour explains why small angle changes close to a vertical line can create large tangent changes. Inverse trigonometric functions work in the other direction, taking a ratio or coordinate and returning a principal angle, but they require quadrant care and are outside this form.

7

Common Mistakes

  • Entering radians as though they were degrees.
  • Swapping x/cosine with y/sine.
  • Ignoring quadrant signs.
  • Treating tangent at 90° as a large number.
  • Assuming a rounded decimal is an exact radical.
  • Forgetting that negative angles rotate clockwise.
  • Using unit-circle output alone to solve an underspecified triangle.
8

Frequently Asked Questions

Are angles entered in degrees?

Yes. Convert radians to degrees before entering them.

Why is x equal to cosine?

On the radius-one circle, horizontal projection is cos θ and vertical projection is sin θ.

When is tangent undefined?

Whenever cosine is zero, at 90° plus any whole multiple of 180°.

Can I enter a negative angle?

Yes. Negative angles rotate clockwise.

What happens above 360 degrees?

The values repeat; subtracting whole multiples of 360° gives a coterminal angle.

Why is an expected zero sometimes shown as zero after rounding?

Floating-point calculations can produce a tiny residual that six-place rounding removes.

Does the page return exact radicals?

No. It displays rounded decimal values.

9

Sources & References