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→Relationships between trigonometric functions and inverse trigonometric functions: The diagram for arc secant and arc cosecant assume that x is positive, and thus adjustments have to be made when x is negative Tags: nowiki added Visual edit |
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=== 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 ''x'', then applying the [[Pythagorean theorem]] and definitions of the trigonometric ratios. Purely algebraic derivations are longer.{{citation needed|date=May 2016}} It's worth noting that for arcsecant and arccosecant, the diagram assumes that x is positive, and thus the result has to be corrected through the use of [[absolute value]]<nowiki/>s and the [[Sign function|signum]] (sgn) operation.
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!<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]]
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!<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{
|[[File:Trigonometric functions and inverse5.svg|150px]]
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