sin 5^{circ} + sin 10^{circ} + sin 15^{circ} | vedclass

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(A) The given series is $S = \sum_{k=1}^{71} \sin(5k^{\circ})$.This is a series of sines in arithmetic progression with $n = 71$ terms,first term $a =

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) The given series is $S = \sum_{k=1}^{71} \sin(5k^{\circ})$.This is a series of sines in arithmetic progression with $n = 71$ terms,first term $a = 5^{\circ}$,and common difference $d = 5^{\circ}$.The formula for the sum is $S = \frac{\sin(n \cdot d/2)}{\sin(d/2)} \cdot \sin\left(\frac{a + l}{2}\right)$,where $l = 355^{\circ}$.$S = \frac{\sin(71 \cdot 5^{\circ}/2)}{\sin(5^{\circ}/2)} \cdot \sin\left(\frac{5^{\circ} + 355^{\circ}}{2}\right)$.$S = \frac{\sin(355^{\circ}/2)}{\sin(5^{\circ}/2)} \cdot \sin(180^{\circ})$.Since $\sin(180^{\circ}) = 0$,the total sum is $0$.

$\sin 5^{\circ} + \sin 10^{\circ} + \sin 15^{\circ} + \dots + \sin 350^{\circ} + \sin 355^{\circ}$ is equal to:

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