Unit Circle Calculator

sin θ = y, cos θ = x, tan θ = y ÷ x

θ

The angle, measured anticlockwise from the positive x-axis. It can be negative, and it can be more than a full turn — only where it lands on the circle matters

cos θ

The x-coordinate of the point the angle lands on. At zero degrees the point is at (1, 0), so the cosine is one; at ninety degrees it is zero

sin θ

The y-coordinate of that same point. At zero degrees it is zero; at ninety degrees the point is at the top of the circle and the sine is one

tan θ

The sine divided by the cosine, which is the slope of the line from the origin to the point. It has no value where the cosine is zero, because that is where the line is vertical

Radius one

Why the coordinates can be read off directly. The point is one unit from the origin, so by Pythagoras the two coordinates always square and add to one — which is the identity cos²θ + sin²θ = 1

Degrees or radians

The dropdown converts whatever you type into degrees before the calculation runs, so the answer is the same either way. A quarter turn is ninety degrees, or half of pi in radians

Four decimal places

How wide the readings are written. Only a handful of readings come out exact — the sine of thirty is a half, the tangent of forty-five is exactly one, and the values sitting on the axes are all whole numbers — and everything else is a value like root two over two that has to be rounded at the end

The unit circle is the table every trigonometry course makes you memorise, because the three trigonometric functions are nothing more than the coordinates of a point on it — and this page is that table with a box you can type any angle into. The immediate uses are the ones where you have an angle and need its components: resolving a force or a velocity into horizontal and vertical parts, which is the sine and cosine pair and nothing else; working out a phase in electronics or a power factor; finding the horizontal reach of something launched at an angle. Surveying and navigation use it whenever a bearing has to be turned into a distance east and a distance north, which is exactly the two coordinates. Graphics and games use it for anything that rotates — a sprite turning, a camera orbiting, a point on a wheel — because the position of a rotating point is the cosine and the sine, and writing that once with this function instead of with case-by-case arithmetic is what makes rotation code short. Then there is the pure-learning use, which is the one this page is really built for: the special angles are worth knowing by heart, and being able to type ninety and see the tangent refuse is a much better way to remember that it has no value there than being told so. The reference table below is the same set of angles, written as exact fractions rather than decimals, because that is the form they are actually used in.

1. An angle of 30 degrees

   1. Rotate 30 degrees anticlockwise from the positive x-axis
   2. The point you land on is (√3/2, 1/2)
   3. The cosine is the x-coordinate: 0.8660
   4. The sine is the y-coordinate: 0.5
   5. The tangent is the ratio: 0.5 ÷ 0.8660 = 0.5774

   The angle the page loads with, and the first row of the table the special angles are usually taught in. Its exact values are root three over two, one half and root three over three — decimals are shown here, exact fractions in the reference table below.

2. An angle of 45 degrees

   1. Rotate 45 degrees anticlockwise, which puts the point on the diagonal
   2. The sine and cosine are equal because the point is at 45 degrees to both axes: √2/2 = 0.7071
   3. The tangent is their ratio, and a number divided by itself is 1

   The only angle between zero and ninety where the sine and cosine are equal, which happens because the point lies on the line y = x. The tangent of exactly 1 is the algebra confirming it — and it is the reason a 45 degree slope is called a one-in-one slope.

3. An angle of 180 degrees

   1. Rotate a half turn, which lands the point on the negative x-axis
   2. The cosine is the x-coordinate: -1
   3. The sine is the y-coordinate: 0
   4. The tangent is zero divided by minus one, which is 0

   A half turn, and the sign that catches people out: the cosine is minus one rather than one, because the point has gone to the opposite side of the circle. The sine comes back as exactly zero here even though the arithmetic inside the page produces a very small number on the way — the reading is rounded at the end, which is the honest way to handle it.

4. An angle of 390 degrees

   1. A full turn brings you back to the start, so 390 degrees is the same as 30
   2. The point is (√3/2, 1/2), exactly as it was for 30 degrees

   The same three readings as the 30 degree example, because going round once more lands on the same point. This is what makes the page usable for angles that are not between zero and a full turn — the wrap-around is handled for you rather than being something to reduce by hand first.

This page takes one angle and returns the sine, cosine and tangent of it. The tangent has no value at ninety degrees or at two hundred and seventy, or at any angle that differs from those by a whole number of turns, because the point has moved to where its x-coordinate is zero and the ratio has nothing to divide by. When that happens the page refuses and explains rather than printing a very large number, and the sine and cosine are perfectly well defined there — one and zero — but this page returns the three together, so it withholds all three rather than two of them; the reference table below carries the exact values for those two. Angles very close to those values are accepted and give very large tangents, which is mathematically correct — a slope close to vertical really is steep — so a near miss is not the same as a hit and the page does not treat it as one. The three readings are pure numbers with no unit, whatever the dropdown says; the dropdown chooses how the angle is entered, not what comes out. Four decimal places is a display width rather than a claim about precision, and most angles do not have exact decimal values at all — the reference table gives the common ones as exact fractions for that reason. Nothing here handles inverse problems: the page turns an angle into coordinates but will not turn coordinates back into an angle, and it gives no arc length, no sector area and no measure of how far round the circle the angle has travelled.

Why does the tangent have no answer at 90 degrees?

Because the tangent is the sine divided by the cosine, and at ninety degrees the cosine is zero. The point has moved to the top of the circle, its x-coordinate is zero, and there is nothing to divide by. The honest description is that the tangent has no value there — not that it is enormous. The sine and cosine are perfectly well defined at that angle — the point is at the top of the circle, so the sine is one and the cosine is zero. This page reports the three together, so it gives none of them rather than two of them: it refuses the angle and says why. The reference table below still carries the exact values for that row.

Can I enter an angle larger than 360 degrees?

Yes, and negative angles too. Only where the angle lands on the circle matters, so 390 degrees gives exactly the same three readings as 30, and minus 30 gives the same as 330. The page reduces the angle for you rather than making you do it by hand first, which matters because bearings and rotations routinely run past a full turn.

Why are the values decimals when the chart at school shows fractions?

Because the results panel is a calculator and the table is a chart. The exact values for the special angles are root two over two, root three over two and their relatives, and those cannot be written as exact decimals — 0.7071 is a rounded version of one of them. The reference table below gives the same angles as exact fractions, which is the form they are worth memorising in.

What is the tangent actually for?

It is a slope. The tangent of the angle is the rise divided by the run of the line from the origin to the point, which is why it turns up in gradients, roof pitches and anything measured as a ratio of vertical to horizontal. The sine and cosine are coordinates; the tangent is the ratio between them, and that difference is the reason only the tangent has holes in its domain.

Does the unit matter?

The dropdown chooses how the angle is entered — degrees, radians or gradians — and nothing else. The three readings are plain numbers with no unit at all, because a coordinate on a circle of radius one has no unit and a ratio of two of them has none either. The page converts your angle to degrees internally, so 0.5236 radians and 30 degrees give identical answers.

Is this the same as a right triangle calculator?

It is the special case of one. In any right triangle the sine is the opposite side over the hypotenuse and the cosine is the adjacent over the hypotenuse; on the unit circle the hypotenuse is one, so the ratios are just the two legs. Every right triangle is a scaled copy of a unit-circle picture, which is why this page's numbers work for triangles of any size.

• Unit Circle — the circle of radius one, why the coordinates of a point on it are the cosine and the sine, and the special angles that have exact values — Wolfram MathWorld (United States)
• Trigonometric Functions — the definitions of sine, cosine and tangent, their domains, and why the tangent is undefined wherever the cosine is zero — Wolfram MathWorld (United States)
• Tangent — the ratio this page divides to get, its period of a half turn rather than a whole one, and its poles at ninety and two hundred and seventy degrees — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); trigonometry of acute angles is part of the compulsory-education mathematics curriculum, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部