Template:EqualOrNegativeIdenticalTrigonometricFunctionsSolutions/doc

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{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions}}

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Equation	if and only if	Solution (for some $k\in \mathbb {Z}$)	Also a solution to
${\phantom {-}}\sin \theta =\sin \varphi$	$\iff$	$\theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	${\phantom {-}}\csc \theta =\csc \varphi$
${\phantom {-}}\cos \theta =\cos \varphi$	$\iff$	$\theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}$	${\phantom {-}}\sec \theta =\sec \varphi$
${\phantom {-}}\tan \theta =\tan \varphi$	$\iff$	$\theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	${\phantom {-}}\cot \theta =\cot \varphi$
$-\sin \theta =\sin \varphi$	$\iff$	$\theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	$-\csc \theta =\csc \varphi$
$-\cos \theta =\cos \varphi$	$\iff$	$\theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}$	$-\sec \theta =\sec \varphi$
$-\tan \theta =\tan \varphi$	$\iff$	$\theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	$-\cot \theta =\cot \varphi$
${\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\&\Updownarrow \\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}$	$\iff$	$\theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	${\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}$

With includeTableDescription

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{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions|includeTableDescription=true}}

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The table below shows how two angles $\theta$ and $\varphi$ must be related if their values under a given trigonometric function are equal or negatives of each other.

Equation	if and only if	Solution (for some $k\in \mathbb {Z}$)	Also a solution to
${\phantom {-}}\sin \theta =\sin \varphi$	$\iff$	$\theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	${\phantom {-}}\csc \theta =\csc \varphi$
${\phantom {-}}\cos \theta =\cos \varphi$	$\iff$	$\theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}$	${\phantom {-}}\sec \theta =\sec \varphi$
${\phantom {-}}\tan \theta =\tan \varphi$	$\iff$	$\theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	${\phantom {-}}\cot \theta =\cot \varphi$
$-\sin \theta =\sin \varphi$	$\iff$	$\theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	$-\csc \theta =\csc \varphi$
$-\cos \theta =\cos \varphi$	$\iff$	$\theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}$	$-\sec \theta =\sec \varphi$
$-\tan \theta =\tan \varphi$	$\iff$	$\theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	$-\cot \theta =\cot \varphi$
${\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\&\Updownarrow \\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}$	$\iff$	$\theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}$	${\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}$

The vertical double arrow $\Updownarrow$ in the last row indicates that $\theta$ and $\varphi$ satisfy $\left|\sin \theta \right|=\left|\sin \varphi \right|$ if and only if they satisfy $\left|\cos \theta \right|=\left|\cos \varphi \right|.$