First slide

Compound angles

Question

$f (x) = \frac{\sin x + \sin 3 x + \sin 5 x + \dots + \sin 15 x}{\cos x + \cos 3 x + \cos 5 x + \dots + \cos 15 x} then the value of f (x) at x = \frac{π}{32}$

Moderate

A

1
B

2
C

3
D

4

Solution

$\begin{array}{l} \sin α + \sin (α + d) + \dots . + \sin (α + (n - 1) d) = = \frac{\sin (\frac{n d}{2})}{\sin \frac{d}{2}} \sin (\frac{α + α + (n - 1) d}{2}) \\ a n d \\ \cos α + \cos (α + d) + \dots . + \cos (α + (n - 1) d) = = \frac{\sin \frac{n d}{2}}{\sin \frac{d}{2}} \cos \frac{(α + α + (n - 1)) d)}{2} \\ ∴ \frac{\sin x + \sin 3 x + \dots . + \sin 15 x}{\cos x + \cos 3 x + \dots . + \cos 15 x} = \frac{\sin (\frac{x + 15 x}{2})}{\cos (\frac{x + 15 x}{2})} = \tan 8 x \end{array}$
$∴ f (\frac{π}{32}) = \tan 8 (\frac{π}{32}) = 1$

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