Real Arccosine/Examples/Sine of Twice Arccosine of x

From ProofWiki
Jump to navigation Jump to search

Example of Use of Real Arccosine

$\map \sin {2 \arccos x}$

can be simplified to:

$2 x \sqrt {1 - x^2}$


Proof

Let $\arccos x = \theta$.

Then:

\(\ds \cos \theta\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds \sin \theta\) \(=\) \(\ds \sqrt {1 - x^2}\) Sum of Squares of Sine and Cosine


Then:

\(\ds \map \sin {2 \arccos x}\) \(=\) \(\ds \sin 2 \theta\)
\(\ds \) \(=\) \(\ds 2 \sin \theta \cos \theta\) Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds 2 x \sqrt {1 - x^2}\)

$\blacksquare$


Sources