Editing Inverse trigonometric functions (section)

=== Relationships between trigonometric functions and inverse trigonometric functions ===

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length <math>x,</math> then applying the [[Pythagorean theorem]] and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that <math>x</math> is positive, and thus the result has to be corrected through the use of [[absolute value]]s and the [[Sign function|signum]] (sgn) operation.

{|class="wikitable"
|-
!<math>\theta</math>
!<math>\sin(\theta)</math>
!<math>\cos(\theta)</math>
!<math>\tan(\theta)</math>
!Diagram
|-
!<math>\arcsin(x)</math>
|<math>\sin(\arcsin(x)) = x </math>
|<math>\cos(\arcsin(x)) = \sqrt{1-x^2}</math>
|<math>\tan(\arcsin(x)) = \frac{x}{\sqrt{1-x^2}}</math>
|[[File:Trigonometric functions and inverse3.svg|150px]]
|-
!<math>\arccos(x)</math>
|<math>\sin(\arccos(x)) = \sqrt{1-x^2}</math>
|<math>\cos(\arccos(x)) = x </math>
|<math>\tan(\arccos(x)) = \frac{\sqrt{1-x^2}}{x}</math>
|[[File:Trigonometric functions and inverse.svg|150px]]
|-
!<math>\arctan(x)</math>
|<math>\sin(\arctan(x)) = \frac{x}{\sqrt{1+x^2}}</math>
|<math>\cos(\arctan(x)) = \frac{1}{\sqrt{1+x^2}}</math>
|<math>\tan(\arctan(x)) = x</math>
|[[File:Trigonometric functions and inverse2.svg|150px]]
|-
!<math>\arccot(x)</math>
|<math>\sin(\arccot(x)) = \frac{1}{\sqrt{1+x^2}}</math>
|<math>\cos(\arccot(x)) = \frac{x}{\sqrt{1+x^2}}</math>
|<math>\tan(\arccot(x)) = \frac{1}{x}</math>
|[[File:Trigonometric functions and inverse4.svg|150px]]
|-
!<math>\arcsec(x)</math>
|<math>\sin(\arcsec(x)) = \frac{\sqrt{x^2-1}}{|x|}</math>
|<math>\cos(\arcsec(x)) = \frac{1}{x}</math>
|<math>\tan(\arcsec(x)) = \sgn(x)\sqrt{x^2-1}</math>
|[[File:Trigonometric functions and inverse6.svg|150px]]
|-
!<math>\arccsc(x)</math>
|<math>\sin(\arccsc(x)) = \frac{1}{x}</math>
|<math>\cos(\arccsc(x)) = \frac{\sqrt{x^2-1}}{|x|}</math>
|<math>\tan(\arccsc(x)) = \frac{\sgn(x)}{\sqrt{x^2-1}}</math>
|[[File:Trigonometric functions and inverse5.svg|150px]]
|-
|}