Real Arccosine/Examples/Sine of Twice Arccosine of x
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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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Inverse ratios