Find angles from trig ratios in seconds — compute arcsin, arccos, and arctan in degrees or radians with exact-value notes. Free, fast, and mobile-friendly.
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Every trigonometry student eventually faces the reverse question: instead of being given an angle and asked for its sine, you are given a ratio and asked for the angle. If sin(θ) = 0.5, what is θ? The answer comes from the inverse trigonometric functions — arcsin, arccos, and arctan — and our free Inverse Trigonometric Functions Calculator evaluates all three instantly, reporting the angle in both degrees and radians with notes when the answer is an exact special value.
These functions are the undo buttons of trigonometry. Where sine maps an angle to a ratio, arcsine maps the ratio back to the angle. They appear everywhere: navigation (what heading gives this course?), engineering (at what angle does this force balance?), physics (launch angle for a projectile), computer graphics (rotating objects toward a target), and surveying (elevation angles from measured distances). Doing them by hand is impractical beyond a few memorized values, which is why a dedicated calculator matters.
One subtlety makes inverse trig richer than it first looks: trigonometric functions are periodic, so infinitely many angles share the same sine. To keep the inverse well-defined, mathematicians restrict each inverse function to a principal range — for example, arcsin always returns an angle between −90° and 90°. The calculator reports principal values, flags exact special angles like 30°, 45°, and 60°, and rejects inputs outside each function’s domain with a clear message.
The three basic inverse trigonometric functions answer three questions. arcsin(x) asks: which angle between −90° and 90° has a sine of x? arccos(x) asks: which angle between 0° and 180° has a cosine of x? arctan(x) asks: which angle between −90° and 90° has a tangent of x? They are written with the prefix “arc” because the angle can be thought of as the arc length on the unit circle that produces the given ratio.
The idea of an inverse is familiar from arithmetic: subtraction undoes addition, and division undoes multiplication. But there is a catch. The sine function is not one-to-one — sin(30°) = sin(150°) = 0.5 — so “the angle whose sine is 0.5” is ambiguous. Mathematicians resolve this by restricting the domain before inverting: sine is cut down to the interval [−90°, 90°], where it rises steadily from −1 to 1 without repeating. The inverse of that restricted sine is arcsin, and it always returns exactly one angle.
Each restriction is chosen to include the most useful angles. Arcsin covers [−90°, 90°] (the right half of the unit circle), arccos covers [0°, 180°] (the upper half), and arctan covers (−90°, 90°). Because tangent shoots off to infinity near ±90°, arctan never actually reaches those endpoints — its graph has horizontal asymptotes, which is why the range is open at both ends. These principal ranges are universal conventions: every calculator, textbook, and programming language (Math.asin, Math.acos, Math.atan) uses them.
The notation deserves a warning. Writing sin⁻¹(x) means the inverse sine, not 1/sin(x). The superscript −1 denotes a function inverse, exactly as f⁻¹ denotes the inverse of a function f. Confusing sin⁻¹(x) with (sin x)⁻¹ = csc(x) is one of the most common errors in trigonometry, so always read the −1 as “undo” rather than “reciprocal.”
θ = arcsin(x) ⟺ sin(θ) = x, −90° ≤ θ ≤ 90° | θ = arccos(x) ⟺ cos(θ) = x, 0° ≤ θ ≤ 180° | θ = arctan(x) ⟺ tan(θ) = x, −90° < θ < 90°
To convert between units: radians = degrees × π/180, and degrees = radians × 180/π. The calculator shows both automatically.
Find arcsin(0.5). We need the angle between −90° and 90° whose sine is 0.5. From the 30°-60°-90° triangle, sin(30°) = 1/2 = 0.5, so arcsin(0.5) = 30°, or π/6 radians (≈ 0.5236). The calculator flags this as an exact value. Note that 150° also has sine 0.5, but it lies outside the principal range, so arcsin does not return it.
Find arccos(−0.5). We need the angle between 0° and 180° whose cosine is −0.5. Cosine is negative in the second quadrant, and cos(120°) = −1/2, so arccos(−0.5) = 120°, or 2π/3 radians (≈ 2.0944). A quick check: cos(120°) = cos(180° − 60°) = −cos(60°) = −0.5. Correct.
Find arctan(1). Tangent equals 1 when opposite equals adjacent, which happens at 45°. So arctan(1) = 45° = π/4 radians (≈ 0.7854). This is one of the most common inverse-trig evaluations in geometry problems — whenever a right triangle is isosceles, the acute angles are arctan(1).
Find arctan(100). Unlike arcsin and arccos, arctan accepts any real input. As x grows, arctan(x) creeps toward 90° without ever reaching it: arctan(100) ≈ 89.427° (≈ 1.5608 radians). This asymptotic behavior is why steep slopes correspond to angles just under 90° — a slope of 100 means a rise of 100 for every 1 unit of run, nearly vertical.
A surveyor knows from a laser rangefinder that a tower is 60 meters tall and she stands 50 meters from its base. The elevation angle is arctan(60/50) = arctan(1.2) ≈ 50.19°. Whenever you know opposite and adjacent sides of a right triangle, arctan turns that ratio into the angle — no protractor needed.
In physics, a ramp problem might state that a 10-newton force pushing into a surface produces a 6-newton normal component, and ask for the ramp’s angle. The normal component equals F·cos(θ), so cos(θ) = 0.6 and θ = arccos(0.6) ≈ 53.13°. This is the famous 3-4-5 triangle angle, and recognizing it lets you solve the whole force diagram — the calculator confirms it instantly.
Navigation software uses a two-argument arctangent (atan2) to convert (x, y) displacements into compass headings, and game developers use arctan to rotate a character toward the mouse pointer: angle = arctan(Δy/Δx). Robotics, satellite dishes, and smartphone screen rotation all depend on the same idea — a measured ratio becomes a direction through an inverse trigonometric function.
The calculator returns the principal value, but equations like sin(θ) = 0.5 have infinitely many solutions. To find them all, use the reference angle — the principal value’s distance from the x-axis — plus the symmetry of the trig functions. For sin(θ) = 0.5, the principal value is 30° (first quadrant); sine is also positive in the second quadrant, giving 180° − 30° = 150°. Adding full rotations: θ = 30° + 360°k or θ = 150° + 360°k for any integer k.
Cosine is positive in quadrants I and IV and negative in II and III. For cos(θ) = −0.5, the principal value 120° sits in quadrant II; the mirror solution is 360° − 120° = 240° in quadrant III. So θ = 120° + 360°k or θ = 240° + 360°k. Tangent has period 180° rather than 360°: tan(θ) = 1 gives θ = 45° + 180°k, covering both the first-quadrant and third-quadrant solutions in one stroke.
This is why the principal range matters so much: it gives you one canonical answer from which every other answer can be generated. In applied problems the context usually selects the right one — a ramp angle is between 0° and 90°, a compass bearing between 0° and 360° — so you rarely need the infinite family, but knowing how to build it separates real understanding from button-pushing.
Find arcsin(−√3/2) ≈ arcsin(−0.8660). Sine is negative in the fourth quadrant within our principal range, and sin(60°) = √3/2, so arcsin(−√3/2) = −60° = −π/3 radians (≈ −1.0472). The negative sign simply mirrors the angle below the x-axis.
Find arccos(0). Cosine equals zero at 90° (and 270°, outside the principal range), so arccos(0) = 90° = π/2 radians (≈ 1.5708). Geometrically, this says the angle between two perpendicular directions has cosine zero — the algebraic statement of perpendicularity.
A useful identity ties the functions together: for any x in [−1, 1], arcsin(x) + arccos(x) = 90° (π/2 radians). This is no coincidence — in a right triangle the two acute angles are complementary, and arcsin and arccos of the same ratio pick out exactly those two angles. Checking that your arcsin and arccos answers sum to 90° is a fast way to verify both at once.
A ladder’s foot is 4 m from a wall and the ladder is 5 m long. What angle does it make with the ground? (Hint: cos(θ) = adjacent/hypotenuse = 4/5, so θ = arccos(0.8).)
Find all angles between 0° and 360° with tan(θ) = √3. (Hint: principal value arctan(√3) = 60°; tangent is also positive in quadrant III.)
Your phone knows which way is up because its accelerometer measures the components of gravity along three axes — and converting those components into a tilt angle is an inverse-trig computation. If the x-axis reads 0.5g while the total is 1g, the tilt from vertical is arcsin(0.5) = 30°. Screen rotation, step counting, and image stabilization all run on this math thousands of times per second.
Game controllers and drones do the same with gyroscopes: a measured angular rate is integrated into an orientation, and inverse trig functions convert the resulting direction vectors back into pitch, roll, and yaw angles the software can display. Every time your map app rotates as you turn, an arctan is deciding which way is forward.
The Inverse Trigonometric Functions Calculator evaluates arcsin, arccos, and arctan for a given value, returning the principal-value angle in degrees and radians, flagging exact special values (30°, 45°, 60°, etc.), enforcing the domains [−1,1] for arcsin/arccos, and explaining each function’s principal range.
arcsin(0.5) = 30° (π/6 radians), because sin(30°) = 0.5 and 30° lies in the principal range [−90°, 90°]. Note that 150° also has sine 0.5, but arcsin returns only the principal value.
There is none — arcsin(x) and sin⁻¹(x) are two notations for the same inverse sine function. Just remember the ⁻¹ means inverse function, not reciprocal: sin⁻¹(x) ≠ 1/sin(x).
Because the sine function itself only outputs values in [−1, 1]. An inverse can only accept outputs of the original function as inputs, so arcsin(2) has no real solution — no angle has a sine of 2.
Use arctan when you know the opposite and adjacent sides (a slope or ratio like rise/run), since tan = opposite/adjacent. It also helps that arctan accepts any real number, while arcsin and arccos are limited to [−1, 1].
Multiply degrees by π/180 to get radians, or radians by 180/π to get degrees. For example, 45° = π/4 ≈ 0.7854 radians. This calculator displays both units for every result.
arcsin: [−90°, 90°]; arccos: [0°, 180°]; arctan: (−90°, 90°). These restricted ranges make each function one-to-one so the inverse returns a single, unambiguous angle.
No finite input makes arctan(x) exactly 90° — the function only approaches 90° asymptotically as x → ∞. For example, arctan(100) ≈ 89.43° and arctan(1,000,000) ≈ 89.99994°.
Start from the principal value α = arcsin(x), then use reference-angle symmetry: the solutions are θ = α + 360°k and θ = (180° − α) + 360°k for any integer k. For cosine use ±360°k symmetry, and for tangent the period is 180°.
Cite this page as: Inverse Trigonometric Functions Calculator. Sigma Calculator. Retrieved October 9, 2026, from https://sigmacalculator.net/math-calculators/inverse-trigonometric-functions-calculator/
Last updated: October 9, 2026