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Inverse trigonometric functions
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==== Equal identical trigonometric functions ==== {{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions|includeTableDescription=true|style=}} ;Set of all solutions to elementary trigonometric equations Thus given a single solution <math>\theta</math> to an elementary trigonometric equation (<math>\sin \theta = y</math> is such an equation, for instance, and because <math>\sin (\arcsin y) = y</math> always holds, <math>\theta := \arcsin y</math> is always a solution), the set of all solutions to it are: {| class="wikitable" style="border: none;" |+ |- ! If <math>\theta</math> solves !! then !! colspan="7" | Set of all solutions (in terms of <math>\theta</math>) |- | style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\sin \theta = y</math> | style="text-align: center;" |then | style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\sin \varphi=y\} =\, </math> | style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,+\, 2</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\pi \Z)</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,\cup\, (-\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>-\pi</math> | style='border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math> <!--------------- END: sin ''θ'' = ''x'' ---------------> |- <!--------------- START: csc ''θ'' = ''r'' ---------------> | style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\csc \theta = r</math> | style="text-align: center;" |then | style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\csc \varphi=r\} =\, </math> | style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,+\, 2</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\pi \Z)</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,\cup\, (-\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>-\pi</math> | style='border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math> <!--------------- END: csc ''θ'' = ''r'' ---------------> |- <!--------------- START: cos ''θ'' = ''x'' ---------------> | style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\cos \theta = x</math> | style="text-align: center;" |then | style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\cos \varphi=x\} =\, </math> | style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,+\, 2</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\pi \Z)</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,\cup\, (-\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' | | style='border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math> <!--------------- END: cos ''θ'' = ''x'' ---------------> |- <!--------------- START: sec ''θ'' = ''r'' ---------------> | style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\sec \theta = r</math> | style="text-align: center;" |then | style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\sec \varphi=r\} =\, </math> | style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>(\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,+\, 2</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\pi \Z)</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,\cup\, (-\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' | | style='border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;' |<math>+ 2 \pi \Z)</math> |- <!--------------- END: sec ''θ'' = ''r'' ---------------> |- <!--------------- START: tan ''θ'' = ''s'' ---------------> | style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\tan \theta = s</math> | style="text-align: center;" |then | style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\tan \varphi=s\} =\, </math> | style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,+\,</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\pi \Z</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' | | style='border-style: solid none solid none; text-align: left; padding: 0;' | | style='border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;' | <!--------------- END: tan ''θ'' = ''s'' ---------------> |- <!--------------- START: cot ''θ'' = ''r'' ---------------> | style="text-align: center; padding: 0.5% 2em 0.5% 2em;" | <math>\;\cot \theta = r</math> | style="text-align: center;" |then | style='border-style: solid none solid none; text-align: left; padding-left: 1em;' |<math>\{\varphi:\cot \varphi=r\} =\, </math> | style='border-style: solid none solid none; text-align: right; padding: 0;' |<math>\theta</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\,+\,</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' |<math>\pi \Z</math> | style='border-style: solid none solid none; text-align: left; padding: 0;' | | style='border-style: solid none solid none; text-align: left; padding: 0;' | | style='border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;' | <!--------------- END: cot ''θ'' = ''r'' ---------------> |}
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