Skip to main content
CalcMax

Reference Angle Calculator

Range: -1,000,000 – 1,000,000

Result

30.00 °

Reference angle

A reference angle calculator takes any angle and returns the reference angle: the acute angle between the angle's terminal side and the nearest part of the horizontal axis, which is always somewhere between 0 and 90 degrees. It is the number trigonometry is built on, because the sine, cosine and tangent of a large angle are the same as those of its reference angle up to a sign, and the signs are the easy part — they come from which quadrant the angle is in. So the standard way to evaluate a trigonometric function of, say, 210 degrees is to notice that its reference angle is 30, that 30 degrees has a sine of one half, and that 210 degrees is in the third quadrant where the sine is negative, giving minus one half. This page does the first half of that and, in its reference table, the second: the quadrant is printed beside each angle. The angle can be anything. It can be negative, in which case it is measured clockwise from the horizontal axis rather than anticlockwise; it can be larger than a full turn, in which case the extra turns are discarded before anything else happens; and it can be a fraction, in which case the answer is a fraction too — the reference angle of 100.25 degrees is 79.75. Whatever goes in, the answer comes out between 0 and 90, which is the property that makes the reference angle useful: it turns an infinite family of angles into a finite table you can learn. The one case that catches people out is the angles that sit exactly on an axis. A reference angle is a distance, not an angle you turn through, and the point at 180 degrees is standing on the horizontal axis, so its reference angle is 0 rather than 180. The same goes for 90 and 270 degrees, which are standing on the vertical axis and so are 90 degrees away from the horizontal one. Those four angles belong to no quadrant at all, and this page says so rather than filing them under the quadrant next door. The table below walks around the circle: one angle from each quadrant, the four axis cases in the middle, and two angles that show the normalisation doing its work — 400 degrees, which is one turn past 40 and gives the same answer as 40, and minus 45 degrees, which is the mirror of 45 and also gives 45.

Reference angles around the circle, including the four that lie on an axis

Angle (°)QuadrantReference angle (°)
30Quadrant I30
135Quadrant II45
210Quadrant III30
300Quadrant IV60
90On an axis90
180On an axis0
400Quadrant I40
-45Quadrant IV45

Eight angles, one from each quadrant on the way round, then the two axis cases, then two that show the wrapping step doing its work. The axis rows are the ones to read carefully. An angle of 180 degrees has a reference angle of 0, not 180: a reference angle is a distance to the horizontal axis and the terminal side at 180 degrees is lying on that axis. The angle of 90 degrees gives 90, because its terminal side stands on the vertical axis and is therefore 90 degrees from the horizontal one — 270 is the same case half a turn further round. Both axis rows belong to no quadrant, which is why the quadrant column reads on an axis rather than naming the quadrant next door. The last two rows are the normalisation: 400 degrees is one turn past 40 and gives the same reference angle as 40 would, and minus 45 is the mirror image of 45 and also gives 45. Note that the answer is never bigger than 90 and never negative, whatever goes into the first column — that is the property the whole idea rests on. Every cell is recomputed from its row when the page is built, in degrees.

Formula

a = ((θ mod 360) + 360) mod 360 a > 180° → a = 360° − a reference angle = min(a, 180° − a)

θ
The angle you enter, in degrees. It can be any real number: positive or negative, less than a full turn or many turns past it, a whole number or a fraction. Negative angles are measured clockwise from the horizontal axis, which is the convention for surveying and for anything that turns the other way
a
The angle folded into the first half turn: the first line wraps the input into the range from 0 up to but not including 360, and the second folds anything past 180 back down. After both, the angle is somewhere between 0 and 180, which is all the information a reference angle needs, because the two halves of the circle are mirror images of each other
Reference angle
The acute angle between the terminal side and the horizontal axis, in degrees, always between 0 and 90. It is a distance rather than a rotation, which is why an angle of 180 degrees has a reference angle of 0: the terminal side is lying on the axis it is being measured from. Printed to two decimals, matching the coterminal angle page, so that an angle given to a quarter of a degree gives an answer at the same precision
mod 360
The remainder after dividing by a full turn, which is what discards whole extra turns. The doubling — add 360, then take the remainder again — is needed because the remainder operator in most programming languages returns a negative result for a negative input, so minus 45 would otherwise come out as minus 45 rather than as 315, and every negative angle would be folded down the wrong branch
min(a, 180° − a)
The distance to the nearest part of the horizontal axis: on a circle of radius one, the angle and the point 180 degrees away from it are the two places the terminal side meets the axis, so the reference angle is whichever of the two is closer. Written as one expression rather than as four quadrant rules it also covers the axis cases without a special branch, which is the whole reason the formula is written this way
Quadrant
Which quarter of the circle the terminal side ends up in once the angle has been wrapped and folded: quadrant one is 0 to 90 degrees, quadrant two is 90 to 180, quadrant three is 180 to 270 and quadrant four is 270 to 360. The four angles that sit exactly on an axis belong to no quadrant, and this page prints them that way rather than filing them with a neighbour

Use this page when you are evaluating a trigonometric function by hand, which is what reference angles are for, and when you need the quadrant along with it. The two go together: the reference angle gives the magnitude of the sine, cosine or tangent, and the quadrant gives the sign, and a page that gave only one of them would leave you doing the other half from memory. The angles where this matters most are the ones you meet in a textbook — 120, 135, 150, 210, 225, 240, 300, 315, 330 — all of which have reference angles drawn from the same short list: 30, 45 and 60. That is the point of the idea. There are infinitely many angles but only a few reference angles worth knowing, so a table of nine angles can be reduced to a table of three, and the rest is bookkeeping about signs. The page is also useful on its own, without any trigonometry, as a way of orienting an angle. If a surveyor gives you a bearing of 315 degrees, the reference angle of 45 tells you it is 45 degrees off the nearest axis, and the quadrant tells you which axis; the same reading answers questions about wind direction, screen coordinates and anything else measured as an angle from a fixed direction. Two cases are worth trying on the page before you need them. Enter 180 and see the answer is 0 — the terminal side is on the axis, and the distance to it is nothing. Enter minus 45 and see the answer is 45 — negative angles are wrapped rather than rejected, and the reference angle of a clockwise angle is the same as that of its anticlockwise mirror image. And if you are comparing this page with the coterminal angle calculator, they take the same input and answer different questions: that page gives you another angle with the same terminal side, this one gives you the distance from that terminal side to the nearest axis. The pair together is what a trigonometry problem usually needs, which is a terminal side identified by both its position and its distance from the axis.

Worked examples

  1. An angle of 210 degrees

    1. Wrap: 210 mod 360 = 210, already inside one turn
    2. Fold: 210 is more than 180, so 360 − 210 = 150
    3. Distance to the nearer axis: min(150, 180 − 150) = min(150, 30) = 30

    The angle the page loads with, and the textbook case: 210 degrees is in the third quadrant, where both the sine and the cosine are negative, and its reference angle is 30. So the sine of 210 degrees is minus the sine of 30, which is minus one half, and the cosine is minus the cosine of 30, which is minus the square root of three over two. The two facts you need for that are on this page and in its table: the magnitude from the reference angle, the signs from the quadrant.

  2. An angle of 135 degrees

    1. Wrap: 135 mod 360 = 135
    2. Fold: 135 is not more than 180, so it stays 135
    3. Distance to the nearer axis: min(135, 180 − 135) = min(135, 45) = 45

    The second quadrant, and the one reference angle worth knowing by heart along with 30 and 60. An angle of 135 degrees is 45 degrees past the vertical axis, so its reference angle is 45 and its sine and cosine are equal in size — the square root of two over two — with the sine positive and the cosine negative. Note the shape of the two steps: the fold is what moves the angle into the second half of the circle, and the distance is what takes it the rest of the way to the axis. Both are needed for anything in quadrants two, three or four.

  3. An angle of 100.25 degrees

    1. Wrap: 100.25 mod 360 = 100.25
    2. Fold: not more than 180, so it stays 100.25
    3. Distance to the nearer axis: 180 − 100.25 = 79.75

    The row that justifies the two decimals. An angle a quarter of a degree past 100 gives a reference angle a quarter of a degree short of 80, and a page that rounded its answers to whole degrees would print 80 and be wrong by a quarter of a degree. Angles in this range are normal in surveying and in astronomy, where they are measured to minutes of arc rather than to whole degrees, and 79.75 is the kind of answer that has to survive the trip through the page intact.

  4. An angle of 180 degrees

    1. Wrap: 180 mod 360 = 180
    2. Fold: 360 − 180 = 180
    3. Distance to the nearer axis: min(180, 180 − 180) = min(180, 0) = 0

    The case that catches people out, and the reason the last step is written as a distance rather than as an angle you turn through. A reference angle measures how far the terminal side is from the horizontal axis, and at 180 degrees the terminal side is lying along that axis, so the distance is zero. The answer is 0 rather than 180, and this angle belongs to no quadrant — the table prints it as being on an axis rather than filing it under the second quadrant or the third, both of which would be wrong.

  5. An angle of minus 45 degrees

    1. Wrap: −45 becomes −45 + 360 = 315
    2. Fold: 315 is more than 180, so 360 − 315 = 45
    3. Distance to the nearer axis: min(45, 180 − 45) = min(45, 135) = 45

    A negative angle, measured clockwise from the horizontal axis, and the row that shows the wrapping step earning its keep: minus 45 is the same terminal side as 315 degrees, which is in the fourth quadrant, and its reference angle is 45 — the same as the positive 45. Clockwise and anticlockwise angles of the same size share a reference angle, and they differ only in the quadrant, which is exactly the division of labour between the two columns of the reference table.

Limitations

The page works in degrees, and the reference angles it prints are in degrees; radians are not accepted in the box and are not printed, so an angle given in radians has to be converted first — multiply by 180 and divide by pi. The answer is always between 0 and 90 inclusive, which means that two very different angles can share one answer: 30, 150, 210 and 330 all have a reference angle of 30, and the reference angle alone cannot tell you which of them you started with. For that you need the quadrant, which this page gives in the table but does not print in the result panel, because a quadrant has no better or worse about it and the badges on this site mean something when they appear. The four angles that lie exactly on an axis — 0, 90, 180 and 270, and their negatives and multiples — have no quadrant at all, and the page says so rather than assigning them to the nearest one; a reader who needs them grouped with a quadrant has to decide which convention their textbook uses. The angle box accepts anything up to a million degrees in either direction, which is far past any real use, but whole extra turns are discarded, so the page is not a tool for counting revolutions. The output is a display width of two decimals rather than a claim about precision, and it is the right width for an angle measured to a quarter or a hundredth of a degree and too generous for one measured to the nearest degree. The page does not evaluate trigonometric functions: it gives the reference angle and, in the table, the quadrant, which are the two ingredients, but the actual value of the sine, cosine or tangent is a separate step and the page does not take it. It also has nothing to say about angles in three dimensions, spherical coordinates or rotations that are not about a fixed axis in a plane, all of which use related ideas but not this one.

Frequently asked questions

How do I find the reference angle of an angle in the third quadrant?
Subtract 180 from it. An angle of 210 degrees in the third quadrant has a reference angle of 30; an angle of 240 has 60; an angle of 225 has 45. The general rule for the whole circle is: subtract 180 from angles between 180 and 270, subtract the angle from 360 for angles between 270 and 360, subtract it from 180 in the second quadrant, and take it as it stands in the first. This page applies all four of those rules through one expression, which is why it also handles the axis cases that the four rules get wrong.
Why is the reference angle of 180 degrees not 180?
Because a reference angle is a distance rather than a rotation. It measures how far the terminal side is from the horizontal axis, and at 180 degrees the terminal side is lying along that axis — the distance is zero. The same reasoning gives 90 for both 90 and 270 degrees, since those two terminal sides are standing on the vertical axis and are 90 degrees away from the horizontal one. All four of those angles lie on an axis and belong to no quadrant, which this page's table prints rather than filing them under the nearest one.
Can the reference angle be negative?
No. The answer is always between 0 and 90 degrees inclusive, whatever goes in. A negative angle is not rejected: minus 45 degrees is measured clockwise from the horizontal axis, which is the same terminal side as 315 degrees, and its reference angle is 45. Clockwise and anticlockwise angles of the same size share a reference angle and differ only in which quadrant they end up in.
What are the reference angles worth memorising?
Three: 30, 45 and 60. Every angle a textbook asks about reduces to one of them — 120, 135, 150, 210, 225, 240, 300, 315 and 330 all come down to one of the three — and the rest of the work is picking the sign from the quadrant. That reduction is the entire point of the reference angle: there are infinitely many angles but only a handful you need to know the trigonometric values of.
What is the difference between a reference angle and a coterminal angle?
A coterminal angle is another angle that ends up in the same place: 400 degrees and 40 degrees have the same terminal side, and the coterminal angle calculator is the page for that. A reference angle is how far that terminal side is from the nearest horizontal axis, always between 0 and 90. They take the same input and answer different questions, and a trigonometry problem usually wants both: the terminal side located, and its distance from the axis.
Does an angle of minus 45 have the same reference angle as 45?
Yes, both are 45. The two angles are mirror images of each other across the horizontal axis, so they are the same distance from it, and they differ only in which side of the axis they lie on — 45 degrees is in the first quadrant and minus 45, or 315, is in the fourth. This is why the answer to a reference angle question is never enough on its own to identify the angle you started with.

References

Related calculators

Coterminal Angle CalculatorAngles coterminal with a given angle, positive or negative.Unit Circle CalculatorSine, cosine and tangent of an angle, read off the unit circle.Arc Length CalculatorThe length of the arc between two points on a circle.Central Angle CalculatorThe angle an arc subtends at the centre of its circle.