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Inverse Trigonometric Functions



Key Terms
Why Restriction Is Necessary
The Arcsine Function
The Arccosine Function
The Arctangent Function
Inverse Reciprocal Functions
Inverse Function Notation
Evaluating Inverse Trigonometric Functions
Compositions of Trigonometric and Inverse Trigonometric Functions
Graphs of Inverse Trigonometric Functions
Summary of the Six Inverse Trigonometric Functions
Inverse Trigonometric Functions FAQ



Recovering Angles from Known Ratios

The six trigonometric functions take an angle and return a number. The inverse trigonometric functions reverse this: they take a number and return an angle. Given that sin⁡(θ)=12\sin(\theta) = \frac{1}{2}, what is θ\theta? The answer is not unique — infinitely many angles satisfy this equation, as the equations page makes clear. But an inverse function must return exactly one value. This forces a restriction: each trigonometric function must be confined to an interval where it is strictly monotonic (always increasing or always decreasing) before an inverse can be defined.

The resulting functions — arcsin⁡\arcsin, arccos⁡\arccos, arctan⁡\arctan, and their reciprocal counterparts — are not merely notational conveniences. They appear as solutions to equations, as building blocks in compositions that simplify using the Pythagorean identity, and as essential tools in calculus (where they arise as antiderivatives of certain algebraic expressions). Their graphs are reflections of the restricted trigonometric graphs over the line y=xy = x, and their domains and ranges are dictated entirely by the properties — specifically the monotonicity — of the original functions.


Key Terms

Inverse Trigonometric Function— returns the angle for a given trigonometric value
Sine— yy-coordinate on the unit circle, restricted to [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] for inversion
Cosine— xx-coordinate on the unit circle, restricted to [0,π][0, \pi] for inversion
Tangent— ratio of sine to cosine, restricted to (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}) for inversion
Trigonometric Ratio— the right-triangle ratios that inverse functions reverse

Formulas Used

Arcsin Plus Arccos— arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2}
Arctan Plus Arccot— arctan⁡x+arccot⁡x=π2\arctan x + \operatorname{arccot} x = \frac{\pi}{2}
Arcsin of Negative— arcsin⁡(−x)=−arcsin⁡x\arcsin(-x) = -\arcsin x
Arccos of Negative— arccos⁡(−x)=π−arccos⁡x\arccos(-x) = \pi - \arccos x
Arctan of Negative— arctan⁡(−x)=−arctan⁡x\arctan(-x) = -\arctan x

See All Trigonometry Definitions →


Why Restriction Is Necessary

A function can have an inverse only if it is one-to-one: each output corresponds to exactly one input. The trigonometric functions, being periodic, are emphatically not one-to-one on their full domains. The equation sin⁡(x)=12\sin(x) = \frac{1}{2} is satisfied by π6\frac{\pi}{6}, 5π6\frac{5\pi}{6}, π6+2π\frac{\pi}{6} + 2\pi, 5π6+2π\frac{5\pi}{6} + 2\pi, and infinitely many others. Defining "sin⁡−1(12)\sin^{-1}\left(\frac{1}{2}\right)" as all of these would not produce a function — a function must assign a single output to each input.

The solution is to restrict each trigonometric function to an interval where it is strictly monotonic — always increasing or always decreasing — and therefore one-to-one. On such an interval, the horizontal line test is passed, and an inverse function exists. The restricted function must still cover the entire range of the original, so that the inverse is defined for every relevant input.

The choice of restriction interval is a convention, universally agreed upon. Different intervals could work (sine is also one-to-one on [π2,3π2]\left[\frac{\pi}{2}, \frac{3\pi}{2}\right], for instance), but the standard choices have been selected for mathematical convenience — they are centered at or near the origin and produce the most natural behavior for compositions and calculus applications.

The Arcsine Function

    The sine function is restricted to [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], where it is strictly increasing and maps onto its full range [−1,1][-1, 1]. The inverse of this restricted sine is called arcsine:

    arcsin⁡(x)=θmeanssin⁡(θ)=xwithθ∈[−π2,π2]\arcsin(x) = \theta \quad \text{means} \quad \sin(\theta) = x \quad \text{with} \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]


    Domain of arcsin⁡\arcsin: [−1,1][-1, 1] — the range of sine. Inputs outside this interval have no corresponding angle.

    Range of arcsin⁡\arcsin: [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] — outputs are always in Quadrant I (for positive inputs), Quadrant IV expressed as negative angles (for negative inputs), or zero.

    Exact values at standard inputs:

  • arcsin⁡(0)=0\arcsin(0) = 0
  • arcsin⁡(12)=π6\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}
  • arcsin⁡(22)=π4\arcsin\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}
  • arcsin⁡(32)=π3\arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}
  • arcsin⁡(1)=π2\arcsin(1) = \frac{\pi}{2}
  • arcsin⁡(−12)=−π6\arcsin\left(-\frac{1}{2}\right) = -\frac{\pi}{6}
  • arcsin⁡(−1)=−π2\arcsin(-1) = -\frac{\pi}{2}

    • The pattern of negative-input values reflects the odd symmetry of arcsine:

    Arcsin of Negative
    arcsin⁡(−x)=−arcsin⁡x\arcsin(-x) = -\arcsin x
    Learn more about this formula: Arcsin of Negative →


    The output is always an angle — a number in radians (or degrees, depending on context). The function answers the question: "What angle between −π2-\frac{\pi}{2} and π2\frac{\pi}{2} has this sine value?"

    The graph of y=arcsin⁡(x)y = \arcsin(x) is obtained by reflecting the restricted sine graph over the line y=xy = x. It is an increasing S-shaped curve, starting at (−1,−π2)(-1, -\frac{\pi}{2}), passing through the origin, and ending at (1,π2)(1, \frac{\pi}{2}).

The Arccosine Function

    The cosine function is restricted to [0,π][0, \pi], where it is strictly decreasing and maps onto [−1,1][-1, 1]. The inverse is arccosine:

    arccos⁡(x)=θmeanscos⁡(θ)=xwithθ∈[0,π]\arccos(x) = \theta \quad \text{means} \quad \cos(\theta) = x \quad \text{with} \quad \theta \in [0, \pi]


    Domain of arccos⁡\arccos: [−1,1][-1, 1].

    Range of arccos⁡\arccos: [0,π][0, \pi] — outputs are always in Quadrant I (for positive inputs) or Quadrant II (for negative inputs).

    Exact values:

  • arccos⁡(1)=0\arccos(1) = 0
  • arccos⁡(32)=π6\arccos\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}
  • arccos⁡(22)=π4\arccos\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}
  • arccos⁡(12)=π3\arccos\left(\frac{1}{2}\right) = \frac{\pi}{3}
  • arccos⁡(0)=π2\arccos(0) = \frac{\pi}{2}
  • arccos⁡(−12)=2π3\arccos\left(-\frac{1}{2}\right) = \frac{2\pi}{3}
  • arccos⁡(−1)=π\arccos(-1) = \pi

    • A fundamental relationship connects arcsine and arccosine:

    Arcsin Plus Arccos
    arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2}
    Learn more about this formula: Arcsin Plus Arccos →


    This is the inverse-function version of the cofunction identity sin⁡θ=cos⁡(π2−θ)\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right). It means knowing one of arcsin⁡(x)\arcsin(x) or arccos⁡(x)\arccos(x) immediately gives the other.

    A related identity captures how arccosine handles negative inputs — unlike arcsine, arccosine is not odd; instead, negating the input supplements the output:

    Arccos of Negative
    arccos⁡(−x)=π−arccos⁡x\arccos(-x) = \pi - \arccos x
    Learn more about this formula: Arccos of Negative →


    The graph of y=arccos⁡(x)y = \arccos(x) is a decreasing curve from (−1,π)(−1, \pi) to (1,0)(1, 0), passing through (0,π2)(0, \frac{\pi}{2}). It is the reflection of the restricted cosine graph over y=xy = x.

The Arctangent Function

    The tangent function is restricted to (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right), where it is strictly increasing and maps onto (−∞,∞)(-\infty, \infty). The inverse is arctangent:

    arctan⁡(x)=θmeanstan⁡(θ)=xwithθ∈(−π2,π2)\arctan(x) = \theta \quad \text{means} \quad \tan(\theta) = x \quad \text{with} \quad \theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)


    Domain of arctan⁡\arctan: (−∞,∞)(-\infty, \infty) — all real numbers, since tangent's range is unbounded.

    Range of arctan⁡\arctan: (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) — an open interval, since tangent approaches but never reaches ±π2\pm\frac{\pi}{2} on its restricted domain.

    Exact values:

  • arctan⁡(0)=0\arctan(0) = 0
  • arctan⁡(33)=π6\arctan\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6}
  • arctan⁡(1)=π4\arctan(1) = \frac{\pi}{4}
  • arctan⁡(3)=π3\arctan(\sqrt{3}) = \frac{\pi}{3}
  • arctan⁡(−1)=−π4\arctan(-1) = -\frac{\pi}{4}
  • arctan⁡(−3)=−π3\arctan(-\sqrt{3}) = -\frac{\pi}{3}

    • Like arcsine, arctangent is an odd function — negating the input negates the output:

    Arctan of Negative
    arctan⁡(−x)=−arctan⁡x\arctan(-x) = -\arctan x
    Learn more about this formula: Arctan of Negative →


    As x→∞x \to \infty, arctan⁡(x)→π2\arctan(x) \to \frac{\pi}{2}. As x→−∞x \to -\infty, arctan⁡(x)→−π2\arctan(x) \to -\frac{\pi}{2}. These are horizontal asymptotes of the arctangent graph — a feature unique among the three primary inverse functions.

    The graph of y=arctan⁡(x)y = \arctan(x) is an increasing S-shaped curve spanning the entire horizontal axis, bounded vertically between −π2-\frac{\pi}{2} and π2\frac{\pi}{2}. It passes through the origin with slope 1 (since ddxarctan⁡(x)∣x=0=1\frac{d}{dx}\arctan(x)\big|_{x=0} = 1) and flattens toward the asymptotes.

    Arctangent is particularly well-behaved: it is defined for all real numbers, it is continuous and differentiable everywhere, and its output is always finite. These properties make it a natural tool in calculus and applied mathematics.

Inverse Reciprocal Functions

    The reciprocal trigonometric functions — cosecant, secant, and cotangent — also have inverses, though they are used less frequently and their conventions vary more across textbooks.

    Arccosecant (arccsc\text{arccsc}): the inverse of cosecant restricted to [−π2,π2]∖{0}\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \setminus \{0\}.

  • (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty)
  • [−π2,0)∪(0,π2]\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]
  • arccsc(x)=arcsin⁡(1x)\text{arccsc}(x) = \arcsin\left(\frac{1}{x}\right)

    • Arcsecant (arcsec\text{arcsec}): the inverse of secant restricted to [0,π]∖{π2}[0, \pi] \setminus \left\{\frac{\pi}{2}\right\}.

  • (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty)
  • [0,π2)∪(π2,π]\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]
  • arcsec(x)=arccos⁡(1x)\text{arcsec}(x) = \arccos\left(\frac{1}{x}\right)

    • Arccotangent (arccot\text{arccot}): the inverse of cotangent restricted to (0,π)(0, \pi).

  • (−∞,∞)(-\infty, \infty)
  • (0,π)(0, \pi)
  • arccot(x)=arctan⁡(1x)\text{arccot}(x) = \arctan\left(\frac{1}{x}\right) for x>0x > 0; requires adjustment for x<0x < 0

    • Arctangent and arccotangent share the same cofunction relationship that arcsine and arccosine do:

    Arctan Plus Arccot
    arctan⁡x+arccot⁡x=π2\arctan x + \operatorname{arccot} x = \frac{\pi}{2}
    Learn more about this formula: Arctan Plus Arccot →


    In practice, these are rarely evaluated directly. When an expression involves arcsec(x)\text{arcsec}(x), it is usually converted to arccos⁡(1x)\arccos\left(\frac{1}{x}\right) for computation. The same applies to arccosecant via arcsine and arccotangent via arctangent. Their primary role is theoretical — they appear in integral formulas in calculus (for example, ∫1xx2−1 dx=arcsec∣x∣+C\int \frac{1}{x\sqrt{x^2 - 1}}\,dx = \text{arcsec}|x| + C) and in certain identity derivations.

Inverse Function Notation

Notation

Inverse Function Notation

The most collided superscript in the subject, the arc-names that dodge it, and the spellings machines use. The table below the entries settles every form side by side.
sin⁡\sin through cot⁡\cot come from the right triangle page; the honest power sin⁡2θ\sin^2\theta that makes sin⁡−1\sin^{-1} treacherous is owned by function notation; the interval marks of the ranges by expressing domains.
sin⁡−1(x)\sin^{-1}(x)
inverse sine of x; sine inverse
An inconsistency by convention: every other exponent on a function name is a power — sin⁡2x=(sin⁡x)2\sin^2 x = (\sin x)^2 — but −1-1 alone means the inverse function. sin⁡−1(x)=arcsin⁡(x)\sin^{-1}(x) = \arcsin(x), an angle out, never 1sin⁡x\frac{1}{\sin x}; the override applies to all six functions.
CasesThe corner mark's fourth career: reciprocals on numbers, inversion on complex numbers and matrices — on functions it flips the map, and only the base tells you which reading applies.
Do not confusecsc⁡x\csc x. By the power pattern, sin⁡−1\sin^{-1} "should" be the reciprocal — writing the reciprocal safely takes brackets, (sin⁡x)−1(\sin x)^{-1}, or the honest 1sin⁡x\frac{1}{\sin x}; exactly the split the table below works through.
arcsin⁡\arcsin · arccos⁡\arccos · arctan⁡\arctan
arcsine, arccosine, arctangent
The collision-free spelling: "the arc whose sine is xx" — on the unit circle the answer-angle is literally an arc length, so the prefix means what it says. Publications wanting zero ambiguity use the arc-names exclusively; both conventions are catalogued among the trigonometry symbols.
CasesThe reciprocal trio extends the pattern — arcsec⁡\operatorname{arcsec}, arccsc⁡\operatorname{arccsc}, arccot⁡\operatorname{arccot} — with conventions that vary by textbook, as Inverse Reciprocal Functions above warns.
Also writtenArcsin⁡\operatorname{Arcsin} — capital-A for the principal branch in older texts, the same capitalization trick complex analysis still uses for its principal argument.
Do not confuseAn arc times a sine. The prefix is part of one operator name — reading "arc · sin" as a product repeats the cancellation error that function abbreviations always invite.
asin⁡\operatorname{asin} · atan2⁡\operatorname{atan2}
a-sine; a-tan-two
The machine spellings: programming languages shorten the arc-names to asin, acos, atan, while calculator buttons keep sin⁡−1\sin^{-1} — two devices, two conventions, one function.
Casesatan2⁡(y,x)\operatorname{atan2}(y, x) is the special one: two arguments, full-quadrant answer — the programming spelling of the principal argument, with yy before xx in the argument list.
Do not confusearctan⁡(y/x)\arctan(y/x). The single-argument version cannot tell opposite quadrants apart — atan2⁡(1,−1)\operatorname{atan2}(1, -1) and arctan⁡(−1)\arctan(-1) differ by π\pi; the two-argument form exists precisely to fix that.
Expression What it means What it returns Unambiguous form
sin⁻¹(x) inverse sine — the angle whose sine is x an angle in [−π/2, π/2] arcsin(x)
(sin x)⁻¹, 1/sin(x) reciprocal of sin(x) a number (the cosecant value) csc(x)

Evaluating Inverse Trigonometric Functions

Evaluating an inverse trigonometric function means answering: "What angle in the restricted range has this function value?"

For standard inputs — the values 0,±12,±22,±32,±10, \pm\frac{1}{2}, \pm\frac{\sqrt{2}}{2}, \pm\frac{\sqrt{3}}{2}, \pm 1 — the answer comes from the unit circle values, filtered through the range restriction.

arcsin⁡(32)=π3\arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} because sin⁡(π3)=32\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} and π3∈[−π2,π2]\frac{\pi}{3} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].

arccos⁡(−22)=3π4\arccos\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} because cos⁡(3π4)=−22\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} and 3π4∈[0,π]\frac{3\pi}{4} \in [0, \pi].

arctan⁡(−1)=−π4\arctan(-1) = -\frac{\pi}{4} because tan⁡(−π4)=−1\tan\left(-\frac{\pi}{4}\right) = -1 and −π4∈(−π2,π2)-\frac{\pi}{4} \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right).

A common trap: arcsin⁡(12)≠5π6\arcsin\left(\frac{1}{2}\right) \neq \frac{5\pi}{6}, even though sin⁡(5π6)=12\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}. The angle 5π6\frac{5\pi}{6} is outside the range [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], so it is not the arcsine output. The correct answer is π6\frac{\pi}{6}.

For non-standard inputs — values like arcsin⁡(0.7)\arcsin(0.7) or arctan⁡(3.5)\arctan(3.5) — a calculator is required. Ensure the calculator is in the correct angle mode (degrees or radians) for the desired output format. Most calculators return radians by default for inverse trigonometric functions.

The standard arcsine and arccosine values across the shared input domain [−1,1][-1, 1] collect into a single reference table, where the cofunction sum identity arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2} is visible row-by-row.
Input x arcsin(x) arccos(x)
−1 −π/2 π
−√3/2 −π/3 5π/6
−√2/2 −π/4 3π/4
−1/2 −π/6 2π/3
0 0 π/2
1/2 π/6 π/3
√2/2 π/4 π/4
√3/2 π/3 π/6
1 π/2 0

Compositions of Trigonometric and Inverse Trigonometric Functions

Compositions like sin⁡(arccos⁡(x))\sin(\arccos(x)) or arcsin⁡(sin⁡(x))\arcsin(\sin(x)) combine a trigonometric function with an inverse. The behavior of these compositions depends on the direction and on whether the input falls within the restricted range.

Direct compositions (function applied to its own inverse):

sin⁡(arcsin⁡(x))=xfor all x∈[−1,1]\sin(\arcsin(x)) = x \quad \text{for all } x \in [-1, 1]

cos⁡(arccos⁡(x))=xfor all x∈[−1,1]\cos(\arccos(x)) = x \quad \text{for all } x \in [-1, 1]

tan⁡(arctan⁡(x))=xfor all x∈(−∞,∞)\tan(\arctan(x)) = x \quad \text{for all } x \in (-\infty, \infty)


These hold universally within the domain — applying a function to its inverse always recovers the input.

Reverse compositions (inverse applied to its own function):

arcsin⁡(sin⁡(x))=xonly if x∈[−π2,π2]\arcsin(\sin(x)) = x \quad \text{only if } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]


If xx is outside this range, the arcsine "folds" the result back into the restricted range. For example, arcsin⁡(sin⁡(5π6))=arcsin⁡(12)=π6\arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right) = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}, not 5π6\frac{5\pi}{6}. The same caution applies to arccos⁡(cos⁡(x))\arccos(\cos(x)) (valid only on [0,π][0, \pi]) and arctan⁡(tan⁡(x))\arctan(\tan(x)) (valid only on (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2})).

Mixed compositions (different functions composed):

sin⁡(arccos⁡(x))\sin(\arccos(x)), cos⁡(arctan⁡(x))\cos(\arctan(x)), tan⁡(arcsin⁡(x))\tan(\arcsin(x)), etc. These are simplified using a right triangle construction:

To evaluate sin⁡(arccos⁡(x))\sin(\arccos(x)): let θ=arccos⁡(x)\theta = \arccos(x), so cos⁡θ=x=x1\cos\theta = x = \frac{x}{1}. Construct a right triangle with adjacent side xx and hypotenuse 11. The opposite side is 1−x2\sqrt{1 - x^2} (by the Pythagorean theorem). Therefore:

sin⁡(arccos⁡(x))=oppositehypotenuse=1−x2\sin(\arccos(x)) = \frac{\text{opposite}}{\text{hypotenuse}} = \sqrt{1 - x^2}


This is valid for x∈[−1,1]x \in [-1, 1], and the result is always non-negative because arccos⁡(x)∈[0,π]\arccos(x) \in [0, \pi], where sine is non-negative.

To evaluate cos⁡(arctan⁡(x))\cos(\arctan(x)): let θ=arctan⁡(x)\theta = \arctan(x), so tan⁡θ=x=x1\tan\theta = x = \frac{x}{1}. Opposite =x= x, adjacent =1= 1, hypotenuse =1+x2= \sqrt{1 + x^2}. Therefore:

cos⁡(arctan⁡(x))=11+x2\cos(\arctan(x)) = \frac{1}{\sqrt{1 + x^2}}


The triangle method works for every mixed composition. It converts the problem from inverse trigonometric territory back to right triangle ratios, using the Pythagorean identity implicitly to find the missing side.

All six mixed compositions — built from the right-triangle method demonstrated above — collect into the reference table below, with the algebraic form and validity domain for each.
Composition Algebraic form Valid for
sin(arccos x) √(1 − x²) x ∈ [−1, 1]
cos(arcsin x) √(1 − x²) x ∈ [−1, 1]
sin(arctan x) x / √(1 + x²) x ∈ ℝ
cos(arctan x) 1 / √(1 + x²) x ∈ ℝ
tan(arcsin x) x / √(1 − x²) x ∈ (−1, 1)
tan(arccos x) √(1 − x²) / x x ∈ [−1, 1], x ≠ 0

Graphs of Inverse Trigonometric Functions

    The graph of each inverse trigonometric function is the reflection of the corresponding restricted trigonometric graph over the line y=xy = x. This reflection swaps the roles of input and output — the domain of the original becomes the range of the inverse, and vice versa.

    y=arcsin⁡(x)y = \arcsin(x) : An increasing S-shaped curve.

  • [−1,1][-1, 1] (horizontal extent)
  • [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] (vertical extent)
  • (−1,−π2)(-1, -\frac{\pi}{2}), (0,0)(0, 0), (1,π2)(1, \frac{\pi}{2})
  • [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}]

    • y=arccos⁡(x)y = \arccos(x) : A decreasing curve.

  • [−1,1][-1, 1]
  • [0,π][0, \pi]
  • (−1,π)(-1, \pi), (0,π2)(0, \frac{\pi}{2}), (1,0)(1, 0)
  • [0,π][0, \pi]

    • y=arctan⁡(x)y = \arctan(x) : An increasing S-shaped curve with horizontal asymptotes.

  • (−∞,∞)(-\infty, \infty)
  • (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)
  • (0,0)(0, 0)
  • y=π2y = \frac{\pi}{2} as x→∞x \to \infty
  • y=−π2y = -\frac{\pi}{2} as x→−∞x \to -\infty
  • R\mathbb{R}

    • All three primary inverse functions are continuous on their domains. Arcsine and arccosine have bounded domains (closed intervals), so their graphs are finite curves with endpoints. Arctangent, with its infinite domain, extends without bound horizontally but is squeezed vertically between the asymptotes — a distinctive shape that appears in probability (the Cauchy distribution), physics (the arctangent potential), and many other contexts.

Summary of the Six Inverse Trigonometric Functions

All six inverse trigonometric functions can be set side-by-side on their definitional attributes — domain, range, monotonicity, the identity that pairs each with another, and the distinctive feature of each function&apos;s graph. The table below makes the structural symmetries of the family immediately visible: the cofunction sum identity holds between arcsin and arccos, and again between arctan and arccot; the three reciprocal inverses share a domain split and connect back to the primary inverses through the 1/x relationship.
Function Domain Range Monotonicity Key identity / relationship Distinctive feature
arcsin [−1, 1] [−π/2, π/2] increasing arcsin x + arccos x = π/2 odd function; passes through origin with slope 1
arccos [−1, 1] [0, π] decreasing arcsin x + arccos x = π/2 neither even nor odd; passes through (0, π/2)
arctan (−∞, ∞) (−π/2, π/2) increasing arctan x + arccot x = π/2 odd; horizontal asymptotes y = ±π/2
arccsc (−∞, −1] ∪ [1, ∞) [−π/2, 0) ∪ (0, π/2] decreasing on each piece arccsc x = arcsin(1/x) undefined for |x| < 1; gap in graph at x = 0
arcsec (−∞, −1] ∪ [1, ∞) [0, π/2) ∪ (π/2, π] increasing on each piece arcsec x = arccos(1/x) undefined for |x| < 1; horizontal asymptote y = π/2
arccot (−∞, ∞) (0, π) decreasing arctan x + arccot x = π/2 horizontal asymptotes y = 0 and y = π

Inverse Trigonometric Functions FAQ

Why do trigonometric functions need restricted domains to have inverses?

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Because a function can only be inverted if each output comes from exactly one input, and the trigonometric functions repeat forever. Sine takes the value 0.5 at infinitely many angles, so an unrestricted inverse would have no way to choose among them. Cutting the domain to one stretch where the function never repeats fixes this.Read more →

What is the difference between sin⁻¹(x) and 1/sin(x)?

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The first is the inverse function, returning the angle whose sine is x. The second is the reciprocal, whose proper name is cosecant. The notation collides because positive exponents do mean powers, so −1 looks like it should too. Writing the reciprocal unambiguously takes brackets, as (sin x)⁻¹, or simply 1 over sin x.Read more →

Why does arcsine only return angles between −π/2 and π/2?

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That is the interval chosen as sine's restricted domain, so it becomes arcsine's range. It works because sine rises steadily across it, hitting every value from −1 to 1 exactly once. Arccosine uses 0 to π for the same reason. Angles outside these ranges have the right sine but are not what the inverse returns.Read more →

How do you evaluate something like sin(arccos(x))?

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Treat the inner part as an angle in a right triangle. If the cosine of that angle is x, label the adjacent side x and the hypotenuse 1, then use the Pythagorean theorem for the third side. Reading the sine off that triangle gives the square root of 1 minus x squared, with no inverse function left.Read more →

Why does arctan sometimes give the wrong quadrant?

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Because it takes a single ratio and cannot distinguish opposite directions: a point in the third quadrant produces the same ratio as one in the first. The result always lands between −π/2 and π/2. The two-argument form atan2, which receives both coordinates separately, exists to resolve exactly this ambiguity.Read more →