Inverse Trigonometric Functions - Revision history

Khanh: /* Solutions to elementary trigonometric equations */

2022-01-15T19:19:20Z

Solutions to elementary trigonometric equations

Khanh: /* Principal values */

2022-01-15T19:09:27Z

Principal values

Khanh: /* Principal values */

2022-01-15T19:04:50Z

Principal values

Khanh: /* Principal values */

2022-01-15T19:00:06Z

Principal values

Khanh at 22:46, 28 October 2021

2021-10-28T22:46:12Z

Khanh at 06:29, 17 September 2021

2021-09-17T06:29:03Z

Khanh: Created page with "In mathematics, the '''inverse trigonometric functions''' (occasionally also called '''arcus functions''', '''antitrigonometric functions''' or '''cyclometric functions''' are..."

2021-09-17T06:25:45Z

Created page with "In mathematics, the '''inverse trigonometric functions''' (occasionally also called '''arcus functions''', '''antitrigonometric functions''' or '''cyclometric functions''' are..."

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-So for example, by using the equality <math>\sin \left(\frac{\pi}{2} - \theta\right) = \cos \theta</math> (found in the table above at row <math>\;\operatorname{func} = \sin\;</math> and column <math>\;\operatorname{func}\left(\frac{\pi}{2} - \theta\right)</math>), the equation <math>\cos \theta = x</math> can be transformed into <math>\sin \left(\frac{\pi}{2} - \theta\right) = x,</math> which allows for the solution to the equation <math>\;\sin \varphi = x\;</math> (where <math>\varphi := \frac{\pi}{2} - \theta</math>) to be used; that solution being:
+So for example, by using the equality <math>\sin \left(\frac{\pi}{2} - \theta\right) = \cos \theta</math>, the equation <math>\cos \theta = x</math> can be transformed into <math>\sin \left(\frac{\pi}{2} - \theta\right) = x,</math> which allows for the solution to the equation <math>\;\sin \varphi = x\;</math> (where <math>\varphi := \frac{\pi}{2} - \theta</math>) to be used; that solution being:
 <math>\varphi  = (-1)^k \arcsin (x) + \pi k \; \text{ for some } k \in \Z,</math>
 which becomes:
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 where using the fact that <math>(-1)^{k} = (-1)^{-k}</math> and substituting <math>h := - k</math> proves that another solution to <math>\;\cos \theta = x\;</math> is:
 <math display="block">\theta ~=~ (-1)^{h+1} \arcsin (x) + \pi h + \frac{\pi}{2} \quad \text{ for some } h \in \Z.</math>
 The substitution <math>\;\arcsin x = \frac{\pi}{2} - \arccos x\;</math> may be used express the right hand side of the above formula in terms of <math>\;\arccos x\;</math> instead of <math>\;\arcsin x.\;</math>
 === Relationships between trigonometric functions and inverse trigonometric functions ===

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-Note: Some authors define the range of arcsecant to be <math> \big( 0 \leq y < \tfrac{\pi}{2} \text{ or } \pi < y \leq \tfrac{3 \pi}{2} \big)</math>, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range <math> \big( 0 \leq y < \tfrac{\pi}{2} \text{ or } \tfrac{\pi}{2} < y \leq \pi \big) </math>, we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math>0 \leq y < \tfrac{\pi}{2},</math> but nonpositive on <math>\tfrac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math> \big(- \pi < y \leq - \tfrac{\pi}{2}</math> \text{ or } 0 < y \leq \tfrac{\pi}{2} \big).</math>
+Note: Some authors define the range of arcsecant to be <math> \big( 0 \leq y < \tfrac{\pi}{2} \text{ or } \pi < y \leq \tfrac{3 \pi}{2} \big)</math>, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range <math> \big( 0 \leq y < \tfrac{\pi}{2} \text{ or } \tfrac{\pi}{2} < y \leq \pi \big) </math>, we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math>0 \leq y < \tfrac{\pi}{2},</math> but nonpositive on <math>\tfrac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math> \big(- \pi < y \leq - \tfrac{\pi}{2} \text{ or } 0 < y \leq \tfrac{\pi}{2} \big).</math>
 If <math>x</math> is allowed to be a complex number, then the range of <math>y</math> applies only to its real part.

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 |-
 |}
-(Note: Some authors define the range of arcsecant to be (<math>0 \leq y < \tfrac{\pi}{2} \text{ or } \pi < y \leq \tfrac{3 \pi}{2}</math>), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range (<math>0 \leq y < \tfrac{\pi}{2} \text{ or } \tfrac{\pi}{2} < y \leq \pi</math>), we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math>0 \leq y < \tfrac{\pi}{2},</math> but nonpositive on <math>\tfrac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math>- \pi < y \leq - \tfrac{\pi}{2}</math> or <math>0 < y \leq \tfrac{\pi}{2}.</math>)
 If <math>x</math> is allowed to be a complex number, then the range of <math>y</math> applies only to its real part.

@@ Line 40: / Line 40: @@
 |-
 |}
-(Note: Some authors define the range of arcsecant to be (<math>0 \leq y < \frac{\pi}{2} \text{ or } \pi < y \leq \frac{3 \pi}{2}</math>), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range (<math>0 \leq y < \frac{\pi}{2} \text{ or } \frac{\pi}{2} < y \leq \pi</math>), we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math>0 \leq y < \frac{\pi}{2},</math> but nonpositive on <math>\frac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math>- \pi < y \leq - \frac{\pi}{2}</math> or <math>0 < y \leq \frac{\pi}{2}.</math>)
+(Note: Some authors define the range of arcsecant to be (<math>0 \leq y < \tfrac{\pi}{2} \text{ or } \pi < y \leq \tfrac{3 \pi}{2}</math>), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math>\tan(\arcsec(x)) = \sqrt{x^2 - 1},</math> whereas with the range (<math>0 \leq y < \tfrac{\pi}{2} \text{ or } \tfrac{\pi}{2} < y \leq \pi</math>), we would have to write <math>\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},</math> since tangent is nonnegative on <math>0 \leq y < \tfrac{\pi}{2},</math> but nonpositive on <math>\tfrac{\pi}{2} < y \leq \pi.</math> For a similar reason, the same authors define the range of arccosecant to be <math>- \pi < y \leq - \tfrac{\pi}{2}</math> or <math>0 < y \leq \tfrac{\pi}{2}.</math>)
 If <math>x</math> is allowed to be a complex number, then the range of <math>y</math> applies only to its real part.

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 * [https://www.youtube.com/watch?v=KbYW9FDm-Zk Derivatives of Inverse Trigonometric Functions], The Organic Chemistry Tutor
-== References==
+== Licensing ==
-# Taczanowski, Stefan (1978-10-01). "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods. ScienceDirect. 155 (3): 543–546. Bibcode:1978NucIM.155..543T. doi:10.1016/0029-554X(78)90541-4.
+Content obtained and/or adapted from:
-# Hazewinkel, Michiel (1994) [1987]. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media. ISBN 978-155608010-4.
+* [https://en.wikipedia.org/wiki/Inverse_trigonometric_functions Inverse trigonometric functions, Wikipedia] under a CC BY-SA license

← Older revision		Revision as of 06:29, 17 September 2021
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−	In mathematics, the '''inverse trigonometric functions''' (occasionally also called '''arcus functions''', '''antitrigonometric functions''' or '''cyclometric functions''' are the [[inverse ~~function]]s~~ of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent (trigonometry)\|tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.	+	In mathematics, the '''inverse trigonometric functions''' (occasionally also called '''arcus functions''', '''antitrigonometric functions''' or '''cyclometric functions''' are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent (trigonometry)\|tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

	==Notation==		==Notation==