First slide

Trigonometric transformations

Question

If $θ_{1}, θ_{2}, θ_{3}, ..... θ_{n}$ are in A.P. then $\frac{\sin θ_{1} + \sin θ_{2} + .... + \sin θ_{n}}{\cos θ_{1} + \cos θ_{2} + ... + \cos θ_{n}} =$

Moderate

A

0
B

$\tan (θ_{1} + θ_{n})$
C

$\tan (\frac{θ_{1} + θ_{n}}{2})$
D

$\tan (\frac{θ_{1} - θ_{n}}{2})$

Solution

$\sin θ_{1} + \sin θ_{2} + ... + \sin θ_{n} =$
$\frac{\sin (\frac{n β}{2})}{\sin \frac{β}{2}} . \sin (\frac{θ_{1} + θ_{n}}{2})$
$\cos θ_{1} + \cos θ_{2} + ... + \cos θ_{n} = \frac{\sin (\frac{n β}{2})}{\sin \frac{β}{2}} . \cos (\frac{θ_{1} + θ_{n}}{2})$
where $β$ is the common difference of angles

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