How this instrument works
Cosine maps an angle to a single number between −1 and 1 — the x-coordinate of a point on the unit circle. Arccos reverses that map: feed it a value x in that same band and it hands back the angle θ whose cosine equals x. Because cosine repeats every full turn, infinitely many angles share a given cosine value, so arccos is built to return exactly one of them — the angle between 0° and 180°, the upper half of the circle, where cosine slides from 1 down to −1 without ever repeating a value.
That restricted half-circle is what makes arccos and arcsin different instruments even though both undo the same unit circle. Arcsin answers with an angle between −90° and 90°, so it can hand back a negative angle; arccos never does — its output always lands between 0° and 180°. The two are locked together by a fixed identity, arccos(x) + arcsin(x) = 90°, because the angle a ray makes with the x-axis and the angle it makes with the y-axis always add to a right angle. Ask for arccos(0.5) and arcsin(0.5) together and the two answers, 60° and 30°, sum to exactly 90° every time.
The name comes straight from Latin: arcus cosinus, the arc whose cosine is known, a reminder that early trigonometry measured angles as arc lengths on a circle of radius one rather than in degrees. At the ends of the domain the function turns steep — its slope is −1 divided by the square root of 1 minus x squared, which climbs toward infinity as x approaches ±1. Moving x from 0.9 to 0.99 barely changes the input, yet the local sensitivity more than triples, from about 2.29 to 7.09 radians of angle per unit of x — a genuinely surprising sting for anyone reading an angle off a cosine measured near the edges of its range.
- Enter your ratio in the x (must be between −1 and 1) field — a value outside that band has no real angle, since cosine itself never produces one.
- Read the answer in the arccos(x) field, which opens showing degrees.
- Switch that field's unit control to radians or turns if your next calculation expects one of those instead.
- To check a right triangle, divide the adjacent side by the hypotenuse — that ratio is cos θ — and enter the result as x.
- Compare the result against 90° minus arcsin of the same x as a quick sanity check, since the two always add to a right angle.
Worked example — arccos(0.5) recovers 60°
Set x to 0.5. The sheet returns θ = 60°, because cos(60°) = 0.5 exactly — arccos has undone cosine and handed back the one angle, out of infinitely many sharing that cosine, that lies in its 0°-to-180° range. Switch the unit control to radians and the same angle reads 1.0471975511965979, matching π ⁄ 3 to the last representable digit; switch again to turns and it reads 0.16666667, one-sixth of a full revolution, since 60° out of 360° is exactly 1 ⁄ 6.
The identity above offers a free check. Feed the same 0.5 into an arcsin calculator and it answers 30°; add that to the 60° found here and the total is 90°, confirming arccos(0.5) + arcsin(0.5) = π ⁄ 2 the way the formula promises. Push x to the edges of its domain and the pattern still holds: arccos(1) returns 0°, arccos(−1) returns 180°, and arccos(0) sits exactly halfway at 90° — three landmark values anyone can check by eye on a unit circle before trusting the sheet with anything less obvious.
Questions
What are the domain and range of arccos?
The domain is −1 to 1, because cosine itself never produces a value outside that interval, so no other input has a real angle to return. The range is 0° to 180° (0 to π radians) — the upper half of the unit circle, the one stretch where cosine steadily falls from 1 to −1 without repeating a value, which is exactly what makes an inverse possible.
How does arccos relate to arcsin?
They are complementary: arccos(x) + arcsin(x) = 90° (π ⁄ 2 radians) for every x from −1 to 1, because the angle a radius makes with the x-axis and the angle it makes with the y-axis always sum to a right angle. The difference that trips people up is range — arcsin can return a negative angle, from −90° to 90°, while arccos never does, always landing between 0° and 180°.
Why does the calculator reject a value like 2 for x?
Because cosine's own output is trapped between −1 and 1 — the x-coordinate of a point on a circle of radius one can never exceed that circle's edge. A value like 2 or −3 has no angle whose cosine equals it in ordinary, real-number trigonometry, so the sheet flags it rather than return a meaningless result.
Where does the name arccos come from?
From Latin arcus cosinus, the arc of the cosine — a holdover from when trigonometry was taught through arc lengths on a circle of radius one rather than through degrees. Asking for arccos(x) is literally asking how long an arc must be, measured in radians, for its cosine to equal x; the degree or turn readout is a later convenience layered on top.
Is arccos(x) the same as 1 divided by cos(x)?
No — that mix-up conflates an inverse function with a reciprocal. 1 ⁄ cos(x) is secant, a different function entirely that can exceed 1 or fall below −1. Arccos(x) instead answers which angle has this cosine, returning a value in degrees, radians, or turns, never a bare ratio.
Why does arccos become so sensitive near x = ±1?
Because its slope, −1 ⁄ √(1 − x²), grows without bound as x approaches either edge of its domain. Moving x from 0.9 to 0.99 changes the local sensitivity from about 2.29 to 7.09 radians of angle per unit of x, so a cosine measured close to −1 or 1 pins down the angle far less precisely than the same measurement error would near x = 0.