Question 635467
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We want to find *[tex \LARGE \cos\,\varphi] if *[tex \LARGE \sin\,\varphi\ =\ \frac{\sqrt{5}}{5}] where *[tex \LARGE \varphi\ =\ \sin^{\small{-1}\LARGE}\,\frac{\sqrt{5}}{5}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin^2\,\varphi\ =\ \frac{1}{5}]


So


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ \cos^2\,\varphi\ =\ \frac{1}{5}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\,\varphi\ =\ \frac{4}{5}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos\,\varphi\ =\ \frac{2\sqrt{5}}{5}]


Substituting back:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos\left(\sin^{\small{-1}\LARGE}\,\frac{\sqrt{5}}{5}\right)\ =\ \pm\frac{2\sqrt{5}}{5}]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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