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Drongoski · Dec 6, 2022

Is answer ??:

$\frac {47}{72} - \sqrt { \frac {\sqrt {119} + 12} {24}}$

=)(= · Dec 6, 2022

soz I dont have the answers but would you be able to explain how you got your answer?

hogzillaAnson · Dec 6, 2022

It's essentially two applications of the double angle formulae. We are in quadrant 4, which means that cosine is + and sine is - (as given). So

$\begin{align*} \sin \theta &= -\frac{5}{12} \\ \cos^2 \theta &= 1- \sin^2 \theta = 1-\frac{25}{144} = \frac{119}{144} \\ \therefore \cos \theta &= \frac{\sqrt{119}}{12}\ [ \cos \theta \geq 0] \\ \implies 2 \cos^2 \frac{\theta}{2} - 1 &= \frac{\sqrt{119}}{12}\ [\text{Double angle formulae}] \\ \implies \cos^2 \frac{\theta}{2} &= \frac{12+\sqrt{119}}{24} \\ \therefore \cos \frac{\theta}{2} &= -\sqrt{\frac{12+\sqrt{119}}{24}}\ \left[\frac{3\pi}{4} < \frac{\theta}{2} < \pi \implies \cos \frac{\theta}{2} < 0\right] \\ \text{Now,}\ \cos 2 \theta &= 1-2 \sin^2 \theta = 1-2 \cdot \frac{25}{144} = \frac{47}{72} \\ \therefore \cos 2\theta + \cos \frac{\theta}{2} &= \frac{47}{72} -\sqrt{\frac{12+\sqrt{119}}{24}}\end{align*}$

=)(= · Dec 7, 2022

ahh i see tysm

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