Question

calculati cos pi/12
sin pi/8
cos pi/8
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Answer 1

Formule pentru semi-unghiuri ale funcțiilor trigonometrice:

$\sin({\frac{x}{2}}) = \sqrt{\frac{1 - \cos(x)}{2}}, x \in [0,\pi)$

$\cos(\frac{x}{2}) = \sqrt{\frac{\cos(x) + 1}{2}}, x \in (-\frac{\pi}{2}, \frac{\pi}{2}]$

$\cos(\frac{\pi}{12}) = \sqrt{\frac{1 + \cos(\frac{\pi}{6})}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \boxed{\frac{\sqrt{2 + \sqrt{3}}}{2}}$

$\sin(\frac{\pi}{8}) = \sqrt{\frac{1 - \cos(\frac{\pi}{4})}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \boxed{\frac{\sqrt{2 - \sqrt{2}}}{2}}$

$\cos{\frac{\pi}{8}} = \boxed{\frac{\sqrt{2 + \sqrt{2}}}{2}}$