First slide

Compound angles

Question

$\cos {7.5}^{\circ} =$

Moderate

A

$\sqrt{\frac{2 + \sqrt{2} + \sqrt{6}}{8}}$
B

$\sqrt{\frac{4 + \sqrt{2} + \sqrt{6}}{8}}$
C

$\frac{2 \sqrt{2} + \sqrt{3} + 1}{2 \sqrt{2}}$
D

$\sqrt{\frac{4 + \sqrt{2} + \sqrt{6}}{4}}$

Solution

Let $θ = {7.5}^{\circ} \Rightarrow 2 θ = 15^{\circ} = 45^{\circ} - 30^{\circ}$
$\begin{array}{l} \Rightarrow \cos 2 θ = \frac{1}{2 \sqrt{2}} (\sqrt{3} + 1) \\ \Rightarrow 2 \cos^{2} θ = \frac{1}{2 \sqrt{2}} (\sqrt{3} + 1) + 1 \\ \Rightarrow \cos^{2} θ = \frac{\sqrt{3} + 1 + 2 \sqrt{2}}{2 \times 2 \sqrt{2}} = \frac{\sqrt{6} + \sqrt{2} + 4}{8} \end{array}$

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