Arcsine of Reciprocal equals Arccosecant
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Theorem
Everywhere that the function is defined:
- $\map \arcsin {\dfrac 1 x} = \arccsc x$
where $\arcsin$ and $\arccsc$ denote arcsine and arccosecant respectively.
Proof
\(\ds \map \arcsin {\frac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \sin y\) | Definition of Real Arcsine | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \csc y\) | Cosecant is Reciprocal of Sine | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \arccsc x\) | \(=\) | \(\ds y\) | Definition of Real Arccosecant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.77$