First slide

Multiple and sub- multiple Angles

Question

Let $f_{n} (θ) = \frac{\cos \frac{θ}{2} + \cos 2 θ + \cos \frac{7 θ}{2} + \dots + \cos (3 n - 2) \frac{θ}{2}}{\sin \frac{θ}{2} + \sin 2 θ + \sin \frac{7 θ}{2} + \dots + \sin (3 n - 2) \frac{θ}{2}}$ then

Moderate

A

$f_{3} (\frac{3 π}{16}) = \sqrt{2} - 1$
B

$f_{5} (\frac{π}{28}) = \sqrt{2} + 1$
C

$f_{7} (\frac{π}{60}) = (2 + \sqrt{3})$
D

none of these

Solution

We have $f_{n} (θ) = \cot ((3 n - 1) \frac{θ}{4})$
$\begin{aligned} ∴ f_{3} (\frac{3 π}{16}) = \cot (8 \times \frac{1}{4} \times \frac{3 π}{16}) = \cot \frac{3 π}{8} = \sqrt{2} - 1 \\ f_{5} (\frac{π}{28}) = \cot (14 \times \frac{1}{4} \times \frac{π}{28}) = \cot \frac{π}{8} = \sqrt{2} + 1 \\ f_{7} (\frac{π}{60}) = \cot (20 \times \frac{1}{4} \times \frac{π}{60}) = \cot \frac{π}{12} = (2 + \sqrt{3}) \end{aligned}$

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