Inverse Cosine Calculator

θ = arccos x x ∈ [−1, 1] θ ∈ [0°, 180°] the other solution in [0°, 360°) is 360° − θ

x

The cosine value you start with, between minus one and one. Outside that band no angle has it, and the page refuses the value rather than approximating

θ

The angle the function returns. Unlike the inverse sine it is never negative, because the range covers the upper half of the circle rather than the right half

360° − θ

The second angle in the circle with the same cosine. Reflections across the horizontal axis share a cosine, which is why the partner is found by subtracting from a full turn here and from a half turn on the inverse sine page

Reach for this page when you have the ratio of the adjacent side to the hypotenuse and want the angle between them — a ladder's angle to the wall, the angle between two vectors worked out from their dot product, a direction recovered from a heading that was stored as a cosine. It is also the function to reach for when a triangle problem gives you three sides and asks for an angle, since the law of cosines produces a cosine and nothing else. 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 upper half and whose answer is signed; the two agree in the first quadrant and part company everywhere else. And if you want the cosine at an angle rather than the angle from a cosine, the cosine page runs the other way.

1. A cosine value of 0.5

   1. The value is inside the band, so there is an angle to find
   2. Sixty degrees is the angle in the range whose cosine is 0.5, so the page reports sixty degrees
   3. In radians that is sixty times pi over a hundred and eighty, which is a third of pi, or 1.0472

   The value the page loads with, and the one that shows how the two inverse pages fit together. The inverse sine of 0.5 is thirty degrees and the inverse cosine of 0.5 is sixty, and thirty plus sixty is ninety — not a coincidence, but the identity that ties the two pages together, true for every value in the band. The same value also shows where the second solution comes from: three hundred degrees has a cosine of 0.5 as well, and it is the reflection of sixty across the horizontal axis, which is why the table's third column reads 300 on this row.

2. A negative cosine value, minus 0.5

   1. Minus 0.5 is inside the band, so the page finds an angle
   2. The angle in the range whose cosine is minus 0.5 is one hundred and twenty degrees
   3. In radians that is two thirds of pi, which is 2.0944

   The row that explains why this page never returns a negative answer. A negative cosine belongs to an obtuse angle, and the range handles that by going past ninety degrees rather than below zero — one hundred and twenty degrees, not minus sixty. The inverse sine page faces the same input and answers the same direction as a signed angle, minus sixty degrees for minus 0.5. Both are correct; the two functions simply partition the circle differently, and the identity still holds: the inverse sine of minus 0.5 is minus thirty, and minus thirty plus one hundred and twenty is ninety.

3. A cosine value of 0.25, which is not a special angle

   1. A quarter is not the cosine of any angle on the usual chart, so the answer will be a decimal
   2. The angle whose cosine is 0.25 is 75.52 degrees
   3. In radians the same angle is 1.3181

   Most inputs land here rather than on a special angle. The row is also the clearest place to see the decrease: a quarter is a small cosine, and a small cosine belongs to an angle near the top of the range rather than near the bottom. Compare the inverse sine page, where a small value gives a small angle. That reversal is not an accident of the two definitions, it is the same reversal you see in the shapes of the two curves, and the tables make it visible in a way the formulas do not.

4. A cosine value of minus 1, at the edge of the band

   1. Minus one is the smallest cosine any angle has, so the answer is the top of the range
   2. One hundred and eighty degrees has a cosine of minus one, and no angle has a smaller one
   3. In radians a half turn is pi, which is 3.1416

   One of the two rows where the second solution is not a second angle. A cosine of minus one belongs to a hundred and eighty degrees and to nothing else in the circle, because the cosine curve turns around at its lowest point and the reflection of that point across the horizontal axis is the point itself. The table shows it: the second and third columns of the first row both read 180. The other end behaves the same way, where a cosine of one belongs to zero degrees alone. Between those two rows every value in the band has a genuine pair, and the pair adds up to a full turn.

Three things this page does not do. First, a value outside the band from minus one to one is refused, because no angle has such a cosine — the same boundary as on the inverse sine page, and for the same reason. Second, the answer is the principal value only. Every cosine between minus one and one belongs to two angles in a full turn and to infinitely many across turns, and the page returns the one in the upper half of the circle; the table's third column has the other one, and nothing here reports how many whole turns away a further copy sits. Third, the radians are printed to four decimals and the degrees to two, so a value that rounds onto a special angle prints as that angle — the inverse cosine of 0.866 prints as thirty degrees, although the exact cosine of thirty degrees is slightly smaller than 0.866.

Why does an inverse cosine never give a negative angle?

Because the range is fixed as the upper half of the circle, from zero to a hundred and eighty degrees. A negative cosine belongs to an obtuse angle, and this range reports it as an angle above ninety rather than folding it below zero the way the inverse sine does. Minus 0.5 has a cosine at one hundred and twenty degrees, not at minus sixty, even though those two angles point in the same direction.

How is this different from the inverse sine?

By the range, and therefore by what happens away from the first quadrant. The inverse sine returns an angle between minus ninety and ninety; the inverse cosine returns one between zero and a hundred and eighty. For every value in the band the two answers add up to ninety degrees, which is the quickest check on either — the inverse cosine of 0.5 is sixty and the inverse sine is thirty. The second solution is found differently too, by subtracting from a full turn here and from a half turn there.

What happens if the value is larger than 1?

The page refuses it. The cosine of an angle stays between minus one and one whatever the angle does, so the range of the inverse function is exactly that band and a value outside it corresponds to no angle at all. This is the same boundary the inverse sine has, and it comes from the same place: a ratio of two sides of a triangle cannot exceed one when the side on the bottom is the longest of the three.

How do I find an angle from a cosine by hand?

Use the values worth remembering — a cosine of 0.5 is sixty degrees, 0.7071 is forty-five, and 0.866 is thirty — and read the rest off a table or a calculator. Then remember the second angle, which is the reflection of the first across the horizontal axis, found by subtracting from three hundred and sixty. This is the page to reach for when a triangle gives you three sides: the law of cosines hands you a cosine and stops there.

Why does the table run backwards?

Because the cosine decreases as the angle grows. Over the range this page reports, from zero to a hundred and eighty degrees, the cosine falls from one to minus one, so a larger input always means a smaller angle. Read the table top to bottom and the answer column goes from a hundred and eighty down to zero. The inverse sine page behaves the other way round, and that difference is the same one you can see in the shapes of the two curves.

Is the inverse cosine the same as one divided by the cosine?

No. The inverse cosine undoes the cosine: give it 0.5 and it returns sixty degrees. One divided by the cosine gives 0.5 the answer 2, which is the secant of sixty degrees — a ratio rather than an angle. The notation makes the confusion easy, because the inverse is often written with a superscript minus one that looks like an exponent and is not one.

• Inverse Cosine — the function this page computes, with its range and the sense in which it decreases — Wolfram MathWorld (United States)
• Inverse Trigonometric Functions — all three inverse functions together, with the ranges that separate them — Wolfram MathWorld (United States)
• Trigonometric Functions — the definitions of sine, cosine and tangent, and the identity that ties the first two together — Wolfram MathWorld (United States)
• Inverse Sine, Cosine, Tangent — the same three functions in plain language, with the range diagrams side by side — Math is Fun (United Kingdom)