First slide

Trigonometric transformations

Question

$\frac{\sin 1^{0} + \sin 2^{0} + \sin 3^{0} + \sin 4^{0}}{\cos 1^{0} + \cos 2^{0} + \cos 3^{0} + \cos 4^{0}} =$

Moderate

A

$\tan {(\frac{5}{2})}^{0}$
B

$\tan 3^{0}$
C

$\tan 4^{0}$
D

$\tan 5^{0}$

Solution

We have $\frac{\sin α + \sin (α + d) + ..... \sin (α + (n - 1) d)}{\cos α + \cos (α + d) + ..... \cos (α + (n - 1) d)} = \frac{\sin (\frac{(α + α + (n - 1)) d}{2})}{\cos (\frac{α + α + (n - 1) d}{2})}$
$∴ \frac{\sin 1^{\circ} + \sin 2^{\circ} + \sin 3^{\circ} + \sin 4^{\circ}}{\cos 1^{\circ} + c o s 2^{\circ} + \cos 3^{\circ} + \cos 4^{\circ}} = \frac{\sin {(\frac{1 + 4}{2})}^{o}}{\cos {(\frac{1 + 4}{2})}^{o}} = \tan {(\frac{5}{2})}^{o}$

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