Template:EqualOrNegativeIdenticalTrigonometricFunctionsSolutions/doc

Usage with no options

Calling

{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions}}

will display

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

With includeTableDescription

Calling

{{EqualOrNegativeIdenticalTrigonometricFunctionsSolutions|includeTableDescription=true}}

will display: The table below shows how two angles $\theta$ and $\phi$ 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 $k\in \mathbb{\mathbb{Z}}$)	Also a solution to
$\phantom{-}\sin \theta =\sin \phi$	$⟺$	$\theta =\phantom{}(-1{)}^k\phi +\phantom{2}\pi k\phantom{+\pi }$	$\phantom{-}\csc \theta =\csc \phi$
$\phantom{-}\cos \theta =\cos \phi$	$⟺$	$\theta =\phantom{-1}\pm \phi +2\pi k\phantom{+\pi }$	$\phantom{-}\sec \theta =\sec \phi$
$\phantom{-}\tan \theta =\tan \phi$	$⟺$	$\theta =\phantom{(-1{)}^{k+1}}\phi +\phantom{2}\pi k\phantom{+\pi }$	$\phantom{-}\cot \theta =\cot \phi$
$-\sin \theta =\sin \phi$	$⟺$	$\theta =(-1{)}^{k+1}\phi +\phantom{2}\pi k\phantom{+\pi }$	$-\csc \theta =\csc \phi$
$-\cos \theta =\cos \phi$	$⟺$	$\theta =\phantom{-1}\pm \phi +2\pi k+\pi \phantom{+\pi }$	$-\sec \theta =\sec \phi$
$-\tan \theta =\tan \phi$	$⟺$	$\theta =\phantom{-1}-\phi +\phantom{2}\pi k\phantom{+\pi }$	$-\cot \theta =\cot \phi$
$\begin{array}{rl}\phantom{-}|\sin \theta |& =|\sin \phi |\\ & \Updownarrow \\ \phantom{-}|\cos \theta |& =|\cos \phi |\end{array}$	$⟺$	$\theta =\phantom{-1}\pm \phi +\phantom{2}\pi k\phantom{+\pi }$	$\begin{array}{rl}\phantom{-}|\tan \theta |& =|\tan \phi |\\ |\csc \theta |& =|\csc \phi |\\ |\sec \theta |& =|\sec \phi |\\ |\cot \theta |& =|\cot \phi |\end{array}$

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