Unit Circle Calculator

What the unit circle is, in one picture

The unit circle is a circle with a radius of 1, drawn with its center at the origin of a graph. That single choice does most of the work in trigonometry, because every point on the circle can be written as (cosine, sine) of the angle that reaches it.

Angles on this circle are measured in radians, and a radian is defined by the circle itself: one radian is the angle you sweep when you walk along the edge for a distance equal to the radius. On a unit circle that means the angle in radians equals the length of the arc you walked. One full lap is the circumference, which is 2π radii, about 6.2832.

Degrees and radians describe the same turn, so the whole circle is 360 degrees and also 6.2832 radians. One radian is 57.2958 degrees, and one degree is 0.017453 radians. The NIST Digital Library of Mathematical Functions states the period of sine and cosine as 2π, which is the same fact written for a computer instead of a protractor.

Read the table once and the pattern shows up: the cosine column falls from 1 to 0 while the sine column rises from 0 to 1, then the signs flip as you keep going around. Nothing is random, and no calculator is needed for these eight angles.

Cosine is the width, sine is the height

Drop a vertical line from a point on the circle to the x axis and you have a right triangle with a hypotenuse of 1. The horizontal leg is the cosine, the vertical leg is the sine, and the tangent is the ratio of the two. That is why the calculator returns a single point for every angle: the point is the answer.

The signs follow the point, not the angle size. At 150 degrees you are in the second quadrant, so the point is (-0.8660, 0.5000): cosine is negative because you are left of the y axis, sine is positive because you are above the x axis, and tangent is -0.5000 divided by 0.8660, which comes out at -0.5774.

A memory aid that survives exams: all functions are positive in quadrant I, sine only in II, tangent only in III, cosine only in IV. Written as a column, it reads A, S, T, C going counterclockwise from the top right.

Students lose more points to signs than to any other part of this topic. If you remember that the point sits left of center in quadrants II and III, the cosine sign can never fool you again.

Degrees, radians, and arc length

Two conversions cover the arithmetic. To go from degrees to radians, multiply by π and divide by 180. To go back, multiply by 180 and divide by π. A 40 degree angle is therefore 40 × π / 180 = 0.6981 radians, and 90 degrees is 90 × π / 180 = 1.5708 radians.

Radians look abstract until you use the arc formulas. On a circle of radius r, an angle of θ radians cuts an arc of length r × θ, and the pie slice behind that arc has an area of 0.5 × r² × θ. On a radius of 6 inches, 40 degrees gives an arc of 6 × 0.6981 = 4.1888 inches and a sector area of 0.5 × 36 × 0.6981 = 12.5664 square inches.

Check the formulas against a full circle and they hold: the circumference is 2 × π × 6 = 37.6991 inches, and the full sector area is 0.5 × 36 × 6.2832 = 113.0973 square inches, which is π × 36. The 2π in the radian measure and the 360 degrees in the degree measure are the same turn, so the answers agree.

Units matter here in a way they do not for sine and cosine. The radian is defined as arc length divided by radius, one length over another length, so it carries no physical unit. The NIST guide to the International System of Units treats the radian and steradian as dimensionless derived units for that reason. Sine and cosine are pure numbers; arc length comes out in whatever length unit you fed in.

Exact values at the angles you get tested on

Most homework and placement tests use the same eight angles, and their exact values are short enough to memorize as a table. The decimals are for checking, the fractions are for writing the answer.

Only two irrational numbers appear: the square root of 2, which is 1.4142, and the square root of 3, which is 1.7321. Halving them gives the 0.7071 and 0.8660 in the table. The tangent column looks less tidy because it is a ratio: 0.5000 divided by 0.8660 is 0.5774, and 0.8660 divided by 0.5000 is 1.7321.

At 90 degrees the tangent has no value. The point is (0, 1), so the ratio y/x would divide by zero. That is not a calculator glitch and not a very large number; it is an angle where the tangent is undefined, and any table that prints a figure there is wrong.

The 45 degree row is worth a second look because sine and cosine are equal. Any point on the line y = x sits at the same distance from both axes, and the only such point on the circle is (0.7071, 0.7071).

Reference angles: reuse the first quadrant

A reference angle is the acute angle between the terminal side and the nearest x axis. It lets you reuse the first-quadrant values you already know, with the sign supplied by the quadrant.

For an angle between 180 and 270 degrees, subtract 180. At 210 degrees the reference angle is 210 - 180 = 30 degrees. Sine takes the sign of y, so sin 210 = -0.5000. Cosine takes the sign of x, so cos 210 = -0.8660. Tangent is positive in quadrant III, so tan 210 = 0.5774, the same magnitude as tan 30.

The rule for the reference angle depends only on the quadrant: in II it is 180 minus the angle, in III it is the angle minus 180, and in IV it is 360 minus the angle. Quadrant I is its own reference angle.

This is the same idea the DLMF identities section writes out as reflection formulas. Working from a reference angle is faster than reaching for a calculator, and it gives the exact form for free.

Tangent is a slope, and where it stops existing

The tangent of an angle is the slope of the line from the origin to the point on the circle. A 45 degree line rises one unit for every unit it runs, so its tangent is 1. A steeper line has a bigger tangent, and a line that tilts the other way has a negative one.

Two common slopes translate neatly into angles. A 2 to 1 slope has a tangent of 2, which is 63.4349 degrees. A roof with a 4 in 12 pitch rises 4 inches for every 12 inches of run, a tangent of 0.3333, which is 18.4349 degrees.

As the angle closes in on 90 degrees, the point's x coordinate shrinks toward zero and the ratio explodes. The tangent of 89 degrees is 57.2900, and the tangent of 89.9 degrees is 572.9572. At exactly 90 degrees the run is zero, the line is vertical, and the tangent is undefined. The same break happens at 270 degrees.

All four are built from the same two coordinates, so none of them needs its own table. The calculator above returns them anyway, because textbook problems often ask for secant or cosecant directly.

Angles past one turn, and negative angles

Angles do not stop at 360 degrees. Going around once more lands you in the same place, so sine, cosine, and tangent repeat every full turn. To handle a large angle, subtract 360 until the angle sits between 0 and 360, then read the circle you already know.

Take 750 degrees. Subtracting two full turns removes 720 degrees and leaves 30, so sin 750 = 0.5000 and cos 750 = 0.8660. A wheel that spins four complete turns, 1440 degrees, ends at the same point as 0 degrees.

Negative angles mean turning clockwise. Minus 30 degrees lands on the same point as 330 degrees, so sin(-30) = -0.5000 and cos(-30) = 0.8660. The cosine stays positive because the point is still right of the y axis.

Radians wrap the same way, just with a different lap length. One lap is 6.2832 radians, so 7.5 radians is 7.5 - 6.2832 = 1.2168 radians, which is 69.72 degrees. The tool above does the subtraction for you and reports the equivalent angle in both units.

Four checks that catch most wrong answers

Four checks catch almost every wrong answer on this topic.

1. Mode. A calculator set to radians reads 30 as 30 radians and returns sin(30) = -0.9880, not 0.5000. If the answer is close to a familiar value but not equal to it, check the mode first.
2. Sign. cos 120 = -0.5000, not 0.5000. The magnitude comes from the reference angle, the sign comes from the quadrant, and mixing them up is the most common error in a timed test.
3. The identity. Sine squared plus cosine squared equals 1 for every angle. At 30 degrees that is 0.25 + 0.75 = 1.00, and at 45 degrees it is 0.5000 + 0.5000 = 1.00. If your two answers do not add up this way, one of them is wrong.
4. Rounding drift. Two times 0.8660 is 1.7320 while the square root of 3 is 1.7321, a difference of 0.0001. Rounding early and then multiplying is how a correct method produces a wrong final digit.

The unit circle is also the fastest way to sanity check a value from a spreadsheet or a programming library. If a function returns a sine outside the range from -1 to 1, the input was almost certainly in the wrong unit.

If you need a full triangle rather than a single angle, the trigonometry calculator takes sides and angles together. For the acute angle between a line and the x axis, the reference angle calculator is the narrower tool.