First slide

Multiple and sub- multiple Angles

Question

If $\sin x + \cos x = \frac{\sqrt{7}}{2}, where x \in (0, \frac{π}{4})$ then $\tan \frac{x}{2}$ is equal to

Moderate

A

$\frac{3 - \sqrt{7}}{3}$
B

$\frac{\sqrt{7} - 2}{3}$
C

$\frac{4 - \sqrt{7}}{3}$
D

none of these

Solution

$\begin{array}{l} \sin x + \cos x = \frac{\sqrt{7}}{2} \\ \Rightarrow \frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}} + \frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}} = \frac{\sqrt{7}}{2} \\ or (\sqrt{7} + 2) \tan^{2} \frac{x}{2} - 4 \tan \frac{x}{2} + (\sqrt{7} - 2) = 0 \end{array}$
$\begin{matrix} \tan \frac{x}{2} = \frac{4 \pm \sqrt{16 - 4 (7 - 4)}}{2 (\sqrt{7} + 2)} = \frac{1}{(\sqrt{7} + 2)} (as \frac{x}{2} < \frac{π}{8}) \\ = \frac{\sqrt{7} - 2}{3} \end{matrix}$

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