What are the exact values of sin, cos and tan on the unit circle?

At 30 degrees, cos is √3/2 (0.8660254), sin is 1/2 (0.5000000) and tan is √3/3 (0.5773503). At 45 degrees cos and sin are both √2/2 (0.7071068) and tan is 1. At 60 degrees cos is 1/2, sin is √3/2 and tan is √3 (1.7320508). Every other common angle repeats one of these three with a sign taken from its quadrant, or sits on an axis where the values are 0, 1 and -1. All 16 are listed in the chart under the tool, and clicking an angle there puts it on the circle.

Why is tan 90 degrees undefined?

Because tan is sin divided by cos, and cos 90 degrees is exactly 0. Dividing by zero gives no number at all, so the tool prints the word undefined rather than a huge figure, at 90, at 270 and at every angle coterminal with them such as -90 and 450. The difference is worth seeing: type the decimal 1.5707963267948966 radians instead, which is near pi/2 but not equal to it, and the page answers cos 6.123234e-17 and tan 1.633124e+16, and says the cosine is within rounding of zero. The header writes that angle as ~90 deg rather than 90, with a tilde (~) marking approximation, because the conversion to degrees lands on 90 while the cosine beside it says the terminal side is not exactly there.

How do I enter radians, or an angle like 5pi/4?

Set the units control to Radians, or just write pi and the page reads it either way. 5pi/4, pi/6, -pi/2, 2*pi/3 and 3/4pi are all understood, as are the π symbol and a trailing word such as 45 deg or 1.2 rad. 5pi/4 is 225 degrees: quadrant III, cos and sin both -0.7071068, tan exactly 1. A plain decimal in radians is answered numerically instead, so 1.2 rad gives 68.7549 degrees and cos 0.3623578. Switching the units control converts the angle in the field rather than reading the same number in the other unit.

What is a coterminal angle, and what happens past 360 degrees?

Coterminal angles share a terminal side, so they share every value. At 225 degrees the page gives -135 as the coterminal angle, one turn back. Type a negative angle and it names the positive partner instead of handing your own angle back, so -90 reads as coterminal with 270, -135 with 225 and -45 with 315. Anything past a full turn is reduced and the turns are counted rather than dropped: 785 degrees reads as 2 full turns and then 65 degrees, with cos 0.4226183, and the stage shows both the angle you typed and the angle it is the same as. Angles up to 1,000,000 degrees in either direction are read.

How does it find the quadrant and the reference angle?

From the angle after the full turns come out. 225 degrees is quadrant III with a reference angle of 45 degrees, 135 degrees is quadrant II with a reference angle of 45, and 300 degrees is quadrant IV with a reference angle of 60. When the terminal side lands on an axis there is no quadrant, and the page says so instead of guessing: at 90 degrees it reports the positive y-axis, and the reference angle is 90.

Can I drag the point, or move it from the keyboard?

Both. Drag anywhere on the circle and the point follows, landing on whole degrees and on a common angle whenever it comes within 4 degrees of one. Focus the circle with the Tab key and the arrow keys move it 1 degree at a time, 15 degrees with Shift held, while Page Up and Page Down jump between the 16 common angles and Home returns to 0. The drawing is also announced as text, so the reading is available without seeing it.

Does it handle negative angles?

Yes, and it draws them the right way round: a negative angle sweeps clockwise from the positive x-axis. At -90 degrees the point is on the negative y-axis, sin is -1, cos is 0 and tan is undefined, and the reading names 270 degrees as the same terminal side. Signs are kept exactly, so -5pi/4 is -225 degrees, in quadrant II.

Can I start with a point, a missing coordinate or all six trig functions?

This page starts with an angle. It reports the point and sin, cos and tan, but has no supplied-point input, inverse-angle solver or missing-coordinate field. Csc, sec and cot are not separate calculated outputs, and the reference-angle output stays in degrees even when the angle input is in radians.

Does this put the unit circle on a physical calculator?

The angle field, draggable circle and common-angle chart work on this browser page. For example, enter 30 degrees to see (√3/2, 1/2), or choose Radians and enter pi/6 for the same point. The page does not install a calculator program or operate a TI, Casio or other physical calculator. Device steps depend on that model and its angle-mode settings.

What happens at an angle between the common chart entries?

The reading is numerical. For example, 22.5 degrees gives cos 0.9238795. The exact/numerical label distinguishes this result from a common-angle surd, and the coordinates and tangent are calculated for that same supplied angle.

Unit Circle Calculator

Enter an angle in degrees or radians, or drag the point, to read its coordinates (cos θ, sin θ), tangent, quadrant, reference angle in degrees and a coterminal angle. The 16 common angles have exact values with decimals beside them; other readings are numerical. Use the chart to load an angle, switch units to convert it, and copy the reading.

225 degreesQuadrant IIIExact values

angle   225 deg = 5pi/4 rad
point   (-sqrt(2)/2, -sqrt(2)/2)

cos -sqrt(2)/2  = -0.7071068
sin -sqrt(2)/2  = -0.7071068
tan  1          =  1.0000000

quadrant III   reference 45 deg   coterminal -135 deg

Angle

Units

cos

-√2/2

sin

-√2/2

tan

Quadrant

III

Angles run from the positive x-axis, counterclockwise when positive and clockwise when negative. 225 degrees is one of the 16 common angles, so cos, sin and tan are exact.

Drag the point, or focus the circle and use the arrow keys: 1 degree a step, 15 with Shift held, Page Up and Page Down to jump between common angles, Home for 0. A dragged point lands on whole degrees, and on a common angle whenever it comes within 4 degrees of one.

30 degreesa common angle
cos √3/2, sin 1/2, tan √3/3
90 degreescos is exactly 0
tan undefined, named rather than infinite
-90 degreesa signed angle
sin -1, coterminal with 270 degrees

The sixteen common angles

Every angle on this chart has exact values, and the tool gives them in this form. Pick one to put it on the circle above.

Radians	cos	sin	tan	Quadrant
0	1	0	0	none (+x axis)
pi/6	√3/2	1/2	√3/3	I
pi/4	√2/2	√2/2	1	I
pi/3	1/2	√3/2	√3	I
pi/2	0	1	undefined	none (+y axis)
2pi/3	-1/2	√3/2	-√3	II
3pi/4	-√2/2	√2/2	-1	II
5pi/6	-√3/2	1/2	-√3/3	II
pi	-1	0	0	none (-x axis)
7pi/6	-√3/2	-1/2	√3/3	III
5pi/4	-√2/2	-√2/2	1	III
4pi/3	-1/2	-√3/2	√3	III
3pi/2	0	-1	undefined	none (-y axis)
5pi/3	1/2	-√3/2	-√3	IV
7pi/4	√2/2	-√2/2	-1	IV
11pi/6	√3/2	-1/2	-√3/3	IV

Accuracy. Exact values at the common rational multiples of π, given in surd form with the decimal beside them, and computed numerically at every other angle. Tangent at an odd multiple of 90 degrees is reported as undefined rather than as a very large number, because the cosine there is exactly zero. Quadrant, reference angle and a coterminal angle are shown for every input you give it.

Common questions

What are the exact values of sin, cos and tan on the unit circle?: At 30 degrees, cos is √3/2 (0.8660254), sin is 1/2 (0.5000000) and tan is √3/3 (0.5773503). At 45 degrees cos and sin are both √2/2 (0.7071068) and tan is 1. At 60 degrees cos is 1/2, sin is √3/2 and tan is √3 (1.7320508). Every other common angle repeats one of these three with a sign taken from its quadrant, or sits on an axis where the values are 0, 1 and -1. All 16 are listed in the chart under the tool, and clicking an angle there puts it on the circle.
Why is tan 90 degrees undefined?: Because tan is sin divided by cos, and cos 90 degrees is exactly 0. Dividing by zero gives no number at all, so the tool prints the word undefined rather than a huge figure, at 90, at 270 and at every angle coterminal with them such as -90 and 450. The difference is worth seeing: type the decimal 1.5707963267948966 radians instead, which is near pi/2 but not equal to it, and the page answers cos 6.123234e-17 and tan 1.633124e+16, and says the cosine is within rounding of zero. The header writes that angle as ~90 deg rather than 90, with a tilde (~) marking approximation, because the conversion to degrees lands on 90 while the cosine beside it says the terminal side is not exactly there.
How do I enter radians, or an angle like 5pi/4?: Set the units control to Radians, or just write pi and the page reads it either way. 5pi/4, pi/6, -pi/2, 2*pi/3 and 3/4pi are all understood, as are the π symbol and a trailing word such as 45 deg or 1.2 rad. 5pi/4 is 225 degrees: quadrant III, cos and sin both -0.7071068, tan exactly 1. A plain decimal in radians is answered numerically instead, so 1.2 rad gives 68.7549 degrees and cos 0.3623578. Switching the units control converts the angle in the field rather than reading the same number in the other unit.
What is a coterminal angle, and what happens past 360 degrees?: Coterminal angles share a terminal side, so they share every value. At 225 degrees the page gives -135 as the coterminal angle, one turn back. Type a negative angle and it names the positive partner instead of handing your own angle back, so -90 reads as coterminal with 270, -135 with 225 and -45 with 315. Anything past a full turn is reduced and the turns are counted rather than dropped: 785 degrees reads as 2 full turns and then 65 degrees, with cos 0.4226183, and the stage shows both the angle you typed and the angle it is the same as. Angles up to 1,000,000 degrees in either direction are read.
How does it find the quadrant and the reference angle?: From the angle after the full turns come out. 225 degrees is quadrant III with a reference angle of 45 degrees, 135 degrees is quadrant II with a reference angle of 45, and 300 degrees is quadrant IV with a reference angle of 60. When the terminal side lands on an axis there is no quadrant, and the page says so instead of guessing: at 90 degrees it reports the positive y-axis, and the reference angle is 90.
Can I drag the point, or move it from the keyboard?: Both. Drag anywhere on the circle and the point follows, landing on whole degrees and on a common angle whenever it comes within 4 degrees of one. Focus the circle with the Tab key and the arrow keys move it 1 degree at a time, 15 degrees with Shift held, while Page Up and Page Down jump between the 16 common angles and Home returns to 0. The drawing is also announced as text, so the reading is available without seeing it.
Does it handle negative angles?: Yes, and it draws them the right way round: a negative angle sweeps clockwise from the positive x-axis. At -90 degrees the point is on the negative y-axis, sin is -1, cos is 0 and tan is undefined, and the reading names 270 degrees as the same terminal side. Signs are kept exactly, so -5pi/4 is -225 degrees, in quadrant II.
Can I start with a point, a missing coordinate or all six trig functions?: This page starts with an angle. It reports the point and sin, cos and tan, but has no supplied-point input, inverse-angle solver or missing-coordinate field. Csc, sec and cot are not separate calculated outputs, and the reference-angle output stays in degrees even when the angle input is in radians.
Does this put the unit circle on a physical calculator?: The angle field, draggable circle and common-angle chart work on this browser page. For example, enter 30 degrees to see (√3/2, 1/2), or choose Radians and enter pi/6 for the same point. The page does not install a calculator program or operate a TI, Casio or other physical calculator. Device steps depend on that model and its angle-mode settings.
What happens at an angle between the common chart entries?: The reading is numerical. For example, 22.5 degrees gives cos 0.9238795. The exact/numerical label distinguishes this result from a common-angle surd, and the coordinates and tangent are calculated for that same supplied angle.

Exact values at the common rational multiples of π, given in surd form with the decimal beside them, and computed numerically at every other angle. Tangent at an odd multiple of 90 degrees is reported as undefined rather than as a very large number, because the cosine there is exactly zero. Quadrant, reference angle and a coterminal angle are shown for every input you give it.