Inverse Trigonometric Functions
Calculate inverse trig functions: arcsin, arccos, arctan, and more.
Quick Examples
Inverse Trigonometric Functions
Inverse trigonometric functions find the angle whose sine, cosine, or tangent equals a given value.
Definitions
- arcsin(x): Angle θ such that sin(θ) = x
- arccos(x): Angle θ such that cos(θ) = x
- arctan(x): Angle θ such that tan(θ) = x
Domain and Range
| Function | Domain | Range |
|---|---|---|
| arcsin(x) | [-1, 1] | [-π/2, π/2] |
| arccos(x) | [-1, 1] | [0, π] |
| arctan(x) | All real | (-π/2, π/2) |
| arcsec(x) | |x| ≥ 1 | [0, π], y ≠ π/2 |
| arccsc(x) | |x| ≥ 1 | [-π/2, π/2], y ≠ 0 |
| arccot(x) | All real | (0, π) |
Key Values
| x | arcsin(x) | arccos(x) | arctan(x) |
|---|---|---|---|
| -1 | -90° | 180° | -45° |
| 0 | 0° | 90° | 0° |
| 1 | 90° | 0° | 45° |
| √3 | — | — | 60° |
| 1/√3 | — | — | 30° |
Worked Examples
- Triangle angle: a right triangle has an opposite side of 2 and a hypotenuse of 4. The ratio is 2/4 = 0.5, so θ = arcsin(0.5) = 30° (π/6 radians ≈ 0.5236).
- Ramp slope: a ramp rises 1 unit over a 12-unit run. θ = arctan(1/12) ≈ 4.76°.
- Game development: a target sits at (8, 6) relative to the origin. The aiming angle is arctan(6/8) = arctan(0.75) ≈ 36.87° (≈ 0.6435 radians).
- From secant: given sec(θ) = 2, use arcsec(2) = arccos(1/2) = 60°. Check: cos(60°) = 0.5, so sec(60°) = 2.
How to Choose the Right Function
| You know | Use | Example situation |
|---|---|---|
| Opposite and hypotenuse | arcsin | Height of a ladder against a wall |
| Adjacent and hypotenuse | arccos | Distance from wall to ladder base |
| Opposite and adjacent | arctan | Rise over run: slopes and ramps |
| Hypotenuse and adjacent | arcsec | Calculus integrals |
| Hypotenuse and opposite | arccsc | Calculus integrals |
Formulas Connecting the Functions
arcsec(x) = arccos(1/x), |x| ≥ 1
arccsc(x) = arcsin(1/x), |x| ≥ 1
arccos(x) = π/2 − arcsin(x)
Principal Values: Why the Calculator Picks One Angle
Infinitely many angles share the same sine — sin(30°) and sin(150°) are both 0.5. To return a single answer, each inverse function is restricted to a principal range: arcsin to [−90°, 90°], arccos to [0°, 180°], arctan to (−90°, 90°). In a triangle problem the principal value is usually what you want, but if the angle could be obtuse, check 180° minus the result as the alternative.
Solving Trigonometric Equations
Solving sin(x) = 1/2 is an inverse-trig problem with a twist: the full solution set is x = π/6 + 2πk or x = 5π/6 + 2πk (for any integer k). The calculator's arcsin(1/2) = 30° gives the first branch; the second branch comes from 180° minus the principal value. The same pattern holds for cosine (360° minus) and tangent (180° periodic).
The atan2 Shortcut
Programming languages provide a two-argument function, atan2(y, x), that returns the correct angle in all four quadrants — something plain arctan(y/x) cannot do because the division destroys the signs of x and y. If you are converting (x, y) coordinates to an angle in code, atan2 is the safe choice.
Notation: sin⁻¹ Is Not 1/sin
The superscript −1 means "inverse function", not reciprocal. arcsin(x) and sin⁻¹(x) are identical notations for the same function; 1/sin(x) is the cosecant, csc(x).
Where Inverse Trig Shows Up
- Construction: a roof with a 6:12 pitch rises at arctan(6/12) ≈ 26.57°.
- Physics: projectile launch angles and the angle of refraction in optics.
- Navigation: bearings between GPS waypoints use arctan.
- Programming: converting (x, y) coordinates to angles for movement and rotation.
Common Mistakes
- Entering impossible values: arcsin(1.5) has no real answer — sine never exceeds 1. Check the ratio calculation.
- Using the wrong function: label the triangle's sides relative to the angle you want before choosing.
- Panicking at a negative angle: arcsin and arctan return negative angles for negative inputs; clockwise from the x-axis is perfectly valid.
- Mixing up units: confirm whether the answer is expected in degrees or radians before finishing.
- Reading sin⁻¹ as 1/sin: the inverse and the reciprocal are different functions.
Special Values Table
| x | arcsin(x) | arccos(x) | arctan(x) |
|---|---|---|---|
| 0 | 0° (0) | 90° (π/2) | 0° (0) |
| 1/2 | 30° (π/6) | 60° (π/3) | — |
| √2/2 | 45° (π/4) | 45° (π/4) | — |
| √3/2 | 60° (π/3) | 30° (π/6) | — |
| 1 | 90° (π/2) | 0° (0) | 45° (π/4) |
| √3 | — | — | 60° (π/3) |
Notice the arcsin and arccos columns are mirror images: arccos(x) = 90° − arcsin(x) for every row.
The Unit-Circle Interpretation
On the unit circle, arcsin answers "given the y-coordinate, find the angle", arccos answers "given the x-coordinate, find the angle", and arctan answers "given the slope y/x, find the angle". This is why arcsin's principal range is [−90°, 90°] (the right half of the circle, where every y appears once) and arccos's is [0°, 180°] (the upper half).
Derivatives of Inverse Trig Functions
| Function | Derivative |
|---|---|
| arcsin(x) | 1/√(1−x²) |
| arccos(x) | −1/√(1−x²) |
| arctan(x) | 1/(1+x²) |
| arccot(x) | −1/(1+x²) |
| arcsec(x) | 1/(|x|√(x²−1)) |
| arccsc(x) | −1/(|x|√(x²−1)) |
Composition Identities
arcsin(sin(x)) = x, only for −π/2 ≤ x ≤ π/2
The round trip is not always the identity: arcsin(sin(150°)) = 30°, because the answer must land in the principal range. This is the single most-tested fact about inverse trig functions.
Why arctan Has Horizontal Asymptotes
As x → ∞, arctan(x) approaches π/2 (90°) but never reaches it — the line y = π/2 is a horizontal asymptote, mirrored by y = −π/2 as x → −∞. This follows from tan approaching ±∞ near ±π/2: the inverse function flattens out exactly where the original function blows up. No other inverse trig function has asymptotes; arcsin and arccos are bounded by their domains.
Inverse Trig in Calculus
These functions appear as antiderivatives of algebraic expressions:
∫ dx/√(1 − x²) = arcsin(x) + C
∫ dx/(x√(x² − 1)) = arcsec|x| + C
Recognizing these patterns is the standard way to crack integrals containing √(1 − x²) or 1 + x² in the denominator.
Calculator Tips
- Match the unit: this calculator offers degrees and radians — pick before computing, not after.
- Sanity-check the range: an arcsin answer outside [−90°, 90°] or an arccos answer outside [0°, 180°] means something went wrong upstream.
- Inputs beyond the domain (like arcsin(2)) have no real answer; check the ratio that produced the input.
Complementary Pair: arcsin + arccos
For any x in [−1, 1], arcsin(x) + arccos(x) = 90° (π/2 radians). Geometrically, the two acute angles of a right triangle sum to 90°, and arcsin and arccos are just those two angles read from opposite sides. This identity lets you convert between the two functions instantly.
A Useful arctan Identity
For x > 0: arctan(x) + arctan(1/x) = π/2. For example, arctan(2) + arctan(0.5) = 90°. (For x < 0 the sum is −π/2 — the sign follows the input.) It is the same complementary-angle relationship, expressed through slopes instead of sides.
Which Branch Does the Calculator Return?
This calculator always returns the principal value: arcsin in [−90°, 90°], arccos in [0°, 180°], arctan in (−90°, 90°), arccot in (0°, 180°), arcsec in [0°, 90°) ∪ (90°, 180°], arccsc in [−90°, 0°) ∪ (0°, 90°]. If your problem allows other quadrants (a second triangle solution, a general trig equation), derive the remaining branches from the principal one using the identities above.