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Inverse Tangent Calculator

Result

45.00 °

Angle

Angle in radians
0.7854 rad

An inverse tangent calculator turns a tangent value into the angle it belongs to, in degrees and in radians. The ratio going in is the one usually written tan. Unlike the other two inverse functions it accepts any real number at all: a slope can be as steep as you like, so there is no band to stay inside, and a tangent of a million is a perfectly good input. The angle that comes back always lies between minus ninety degrees and ninety, and the two ends of that interval are the interesting part, because neither of them is ever reached. A tangent of ninety degrees does not exist — the ratio would put the adjacent side at zero and divide by it. So however large the value you enter, the answer climbs towards ninety and stops short of it, and the page prints ninety degrees for anything big enough that the difference falls below the last decimal place it shows. That is a rounding artefact rather than an answer, and it is worth knowing about before it happens. The rest of the page follows from one more property. The tangent repeats every half turn rather than every full one, so a tangent of one belongs to forty-five degrees and to two hundred and twenty-five, and those two are a half turn apart rather than reflections of each other. The table below shows nine values with their angle, their radians, and the partner angle half a turn away. Two neighbours are worth a look. If the ratio you hold is the opposite side over the hypotenuse, that is the inverse sine, which lives inside the band from minus one to one and answers in the right half of the circle. And if it is the adjacent over the hypotenuse, that is the inverse cosine, whose range is the upper half of the circle and whose answers are never negative.

Nine tangent values with the principal angle and the half-turn partner

tan θθ (degrees)other θ (degrees)θ (radians)
-3-71.57108.43-1.249
-1.7321-60120-1.0472
-1-45135-0.7854
-0.5774-30150-0.5236
001800
0.5774302100.5236
1452250.7854
1.7321602401.0472
371.57251.571.249

This is the only table of the three inverse pages with no row where the two angle columns agree, and the reason is worth a moment. On the inverse sine page and the inverse cosine page the second solution is a reflection, so at the two turning points of the curve the partner is the angle itself and the columns print the same number; the rows of those tables that read ninety with ninety are that collapse. The tangent has no turning points and no bound: it climbs from minus infinity to plus infinity over a single half turn, and the two angle columns are always half a turn apart. So every row here differs by a hundred and eighty, and there is no row at the ends to collapse because the ends are not in the table — the range is open, and the angles climb towards ninety and minus ninety without ever being them. Read the first and last rows together: three and minus three give angles that are mirror images of each other, seventy-one point five seven above zero and the same below it, and the two partner columns mirror each other in the same way about the middle row.

Formula

θ = arctan x x ∈ ℝ θ ∈ (−90°, 90°) the other solution in [0°, 360°) is θ + 180°

x
The tangent value, which may be any real number. Nothing is refused here, because the tangent takes every real value somewhere on the circle
θ
The angle the function returns, always strictly between minus ninety degrees and ninety. Both ends are excluded, because the tangent has no value at either
θ + 180°
The second angle in the circle with the same tangent. The tangent repeats every half turn, so the partner is found by adding rather than by reflecting across an axis

Reach for this page whenever the ratio you have is a slope — rise over run, the gradient of a road or a roof, the direction of a vector in the plane. It is also the inverse function to reach for when the two sides you know are the ones that meet at the right angle, since the tangent of an angle is the opposite side over the adjacent one, and it is the one that never turns down an input for being too large. Two neighbours matter. If the ratio you hold is the opposite side over the hypotenuse, that is the inverse sine, whose range covers the right half of the circle instead of the middle band. And if what you want is the tangent at an angle rather than the angle from a tangent, the tangent page runs the other way, and its own page explains why the ratio blows up at a right angle.

Worked examples

  1. A tangent value of 1

    1. The value is accepted whatever it is, so there is an angle to find
    2. Forty-five degrees is the angle in the range whose tangent is 1, so the page reports forty-five degrees
    3. In radians that is a quarter of pi, which is 0.7854

    The value the page loads with, and the smallest example of the half-turn partner. A tangent of 1 also belongs to two hundred and twenty-five degrees, which is forty-five plus half a turn — the table's third column reads 225 on that row. Note what the partner is not: it is not a reflection of forty-five across either axis, the way the second solution is on the inverse sine and inverse cosine pages. The tangent simply repeats itself half a turn later, and that is the same statement as the tangent having a period of a hundred and eighty degrees rather than three hundred and sixty.

  2. A negative tangent value, minus 1

    1. The value is accepted, and it is negative, so the answer will be too
    2. The angle in the range whose tangent is minus 1 is minus forty-five degrees
    3. The radians follow the sign: minus 0.7854

    The signed convention, as on the inverse sine page and unlike the inverse cosine. A negative slope gives a negative angle, because the range is the band from minus ninety to ninety and half of it is below zero. The partner here is a hundred and thirty-five degrees, which is minus forty-five plus a half turn, and it is also the reflection of forty-five across the vertical axis — but that coincidence belongs to this value rather than to the page, and it stops holding as soon as the value is not exactly minus one.

  3. A tangent value of 0.5, which is not a special angle

    1. A half is not the tangent of any angle on the usual chart, so the answer will be a decimal
    2. The angle whose tangent is 0.5 is 26.57 degrees
    3. In radians the same angle is 0.4636

    Most slopes land on a row like this rather than on a special angle. It is also the clearest place to see the shape of the answer: as the value grows the angle climbs towards ninety and never touches it, and as the value shrinks towards zero the angle falls towards zero. The tangent of a small angle is very close to the angle itself in radians — 0.4636 against 0.5 — which is the same small angle approximation that shows up on the inverse sine page, and it is looser here than it is there.

  4. A tangent value of a million, at the edge of the range

    1. A million is an ordinary input here, since no bound applies
    2. The angle whose tangent is a million is 89.99994 degrees, a hair below ninety
    3. Two decimal places are not enough to show the difference, so the page prints ninety degrees

    The row the page exists to warn about. Ninety degrees is not in the range and never will be — the tangent has no value there at all — but an input this large lands close enough that the printed answer rounds up to it. The printed figure is honest about what it is: the angle is ninety degrees to the precision shown. What would be wrong is reading it as an exact value, or worse, as the tangent of ninety degrees being a million rather than undefined. Push the input further, to a hundred million, and the true angle moves closer to ninety without the reading changing.

Limitations

Three things this page does not do. First, the endpoints of the range are never returned. Ninety degrees and minus ninety are excluded, because the tangent has no value there, so the answer approaches them without arriving; for inputs large enough that the difference falls below two decimal places the page prints the endpoint anyway, and that reading is a rounded one rather than an exact one. Second, the answer is the principal value only. The tangent repeats every half turn, so every value on this page belongs to two angles in a full turn and to infinitely many across turns, and the page returns the one strictly between minus ninety and ninety; the table's third column has the partner, which is half a turn away. Third, the radians are printed to four decimals and the degrees to two, so a value that lands near a special angle after rounding will print as that angle — the inverse tangent of 1.7321 prints as sixty degrees, although the exact tangent of sixty degrees is slightly larger.

Frequently asked questions

Why is any value allowed here when the other two refuse values above 1?
Because the tangent has no upper bound. The inverse sine and inverse cosine take a ratio of two sides with the longest side on the bottom, so their input cannot exceed one; the tangent is the opposite side over the adjacent one, and neither of those is the longest side, so the ratio can be as large as the triangle is steep. A tangent of a million is the slope of a wall a million times taller than it is wide.
Why does a very large value give exactly 90 degrees?
It does not, strictly speaking — it gives an angle just below ninety that rounds to it. Ninety degrees is not in the range, because the tangent has no value there: the ratio would divide by zero. As the input grows the answer climbs towards ninety and never arrives, so for anything above roughly ten thousand the printed answer is ninety degrees to the two decimal places shown. The reading is correct at that precision and wrong if you treat it as exact.
Why is the second angle half a turn away rather than a reflection?
Because the tangent repeats every hundred and eighty degrees rather than every three hundred and sixty. Adding half a turn to an angle leaves both the opposite and the adjacent side flipped in sign, so their ratio is unchanged. The inverse sine and inverse cosine pages find their second solution by reflecting across an axis, since sine and cosine are shared by mirror images; the tangent is not, so its partner comes from adding rather than subtracting.
How do I find an angle from a tangent by hand?
Remember that a tangent of 1 is forty-five degrees and a tangent of 1.7321 is sixty, work out the rest from a table or a calculator, and then add a hundred and eighty degrees if you want the other angle in the circle. This is the inverse function to use for a slope — rise over run — and the one to use when the two sides you know are the two that meet at the right angle.
Why is the answer negative for a negative value?
Because the range runs from minus ninety degrees to ninety, and half of it lies below zero. A negative tangent means the opposite and adjacent sides have opposite signs, which is a slope going down rather than up, and the page reports it as the sign of the input suggests. The partner angle, half a turn away, is a positive one — minus forty-five and a hundred and thirty-five are the same slope in opposite directions.
Is the inverse tangent the same as one divided by the tangent?
No. The inverse tangent undoes the tangent: give it 1 and it returns forty-five degrees. One divided by the tangent gives 1 the answer 1, which is the cotangent of forty-five degrees — a ratio rather than an angle. The two coincide only at forty-five degrees, which is exactly the kind of coincidence that makes the confusion so persistent.

References

Related calculators

Cotangent CalculatorCotangent of an angle in degrees or radians.Trigonometry CalculatorSine, cosine and tangent of any angle, in degrees or radians.Inverse Sine CalculatorAngle from a sine value, in degrees or radians.Inverse Cosine CalculatorAngle from a cosine value, in degrees or radians.