y=arccsc⁡xy = \operatorname{arccsc} x

The Inverse Cosecant y=arccsc⁡xy = \operatorname{arccsc} x

arccsc⁡x\operatorname{arccsc} x, the inverse cosecant, inverts csc⁡x=1sin⁡x\csc x = \dfrac{1}{\sin x}1. The usual choice of range is [−π2,π2]\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right], matching arcsin⁡\arcsin, and it pairs with the inverse secant.

Definition and closed form

The equation csc⁡y=x\csc y = x is the same as sin⁡y=1x\sin y = \dfrac{1}{x}, so the function can be written in closed form.

arccsc⁡x=arcsin⁡1x\operatorname{arccsc} x = \arcsin\frac{1}{x}

Since arcsin⁡\arcsin is defined on [−1,1][-1, 1], the domain here is ∣x∣≥1|x| \geq 1.

Domain and range

  • The domain is x≤−1x \leq -1 or x≥1x \geq 1
  • The range is [−π2,π2]\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right] with 00 removed
  • At x=1x = 1 the value is π2\dfrac{\pi}{2}, and at x=−1x = -1 it is −π2-\dfrac{\pi}{2}

Attaining 00 would require 1x=0\dfrac{1}{x} = 0, so that value is out of reach.

Symmetry and asymptote

Both arcsin⁡\arcsin and 1x\dfrac{1}{x} are odd, so their composition is odd and the graph has rotational symmetry about the origin, though the origin itself lies outside the domain.

As x→±∞x \to \pm\infty we get y→0y \to 0, making the xx-axis a horizontal asymptote, approached from above on the right and from below on the left. Since 00 is never attained, the two branches run alongside the axis without ever meeting it.

Notable values

xx1x\dfrac{1}{x}arccsc⁡x\operatorname{arccsc} x
−1-1−1-1−π2-\dfrac{\pi}{2}
−2-2−12-\dfrac{1}{2}−π6-\dfrac{\pi}{6}
1111π2\dfrac{\pi}{2}
2\sqrt{2}22\dfrac{\sqrt{2}}{2}π4\dfrac{\pi}{4}
2212\dfrac{1}{2}π6\dfrac{\pi}{6}

Monotonicity and tangents

The derivative is as follows.

ddxarccsc⁡x=−1∣x∣x2−1\frac{d}{dx}\operatorname{arccsc} x = -\frac{1}{|x|\sqrt{x^{2}-1}}

It is negative throughout the domain, so the function decreases on both branches. It differs from the derivative of arcsec⁡\operatorname{arcsec} only in sign, inherited directly from the relationship between arccos⁡\arccos and arcsin⁡\arcsin. As ∣x∣→1|x| \to 1 the derivative diverges, so the tangents at the endpoints (1,π2)\left( 1, \dfrac{\pi}{2} \right) and (−1,−π2)\left( -1, -\dfrac{\pi}{2} \right) are vertical.

Relation to the inverse secant

arcsec⁡x+arccsc⁡x=π2\operatorname{arcsec} x + \operatorname{arccsc} x = \frac{\pi}{2}

This is arccos⁡u+arcsin⁡u=π2\arccos u + \arcsin u = \dfrac{\pi}{2} with u=1xu = \dfrac{1}{x}. Where arcsec⁡\operatorname{arcsec} climbs toward π2\dfrac{\pi}{2} from below, this one falls toward 00 from above, and the two graphs are reflections of each other in the line y=π4y = \dfrac{\pi}{4}.

The full family

With this the six inverse trigonometric functions are complete. The range of each trigonometric function passes straight over to become the domain of its inverse.

FunctionDomainRange
arcsin⁡x\arcsin x[−1,1][-1, 1][−π2,π2]\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right]
arccos⁡x\arccos x[−1,1][-1, 1][0,π][0, \pi]
arctan⁡x\arctan xall real numbers(−π2,π2)\left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right)
arccot⁡x\operatorname{arccot} xall real numbers(0,π)(0, \pi)
arcsec⁡x\operatorname{arcsec} x∣x∣≥1|x| \geq 1[0,π][0, \pi] without π2\dfrac{\pi}{2}
arccsc⁡x\operatorname{arccsc} x∣x∣≥1|x| \geq 1[−π2,π2]\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right] without 00

Of these, only arccot⁡\operatorname{arccot} has a genuinely contested principal value; the other five are settled by broad agreement.

  1. Inverse trigonometric functions, Wikipedia