Find the value of $sin (10 °) + sin (20 °) + sin (30 °) + sin (40 °) + . . . . . . + sin (360 °)$

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Solution

The given angles $10 °, 20 °, 30 °, . . . . ., 360 °$ are in AP with total number of terms 36 and common difference $10^{\circ} .$
Now, let's take the later $18$ terms i.e. from terms $sin 190^{\circ}$ to $sin 360^{\circ}$
$\Rightarrow sin 190^{\circ} + sin 200^{\circ} + sin 210^{\circ} + \dots + sin 350^{\circ} + sin 360^{\circ}$
$\Rightarrow sin (180^{\circ} + 10^{\circ}) + sin (180^{\circ} + 20^{\circ}) + sin (180^{\circ} + 30^{\circ}) + \dots + sin (180^{\circ} + 170^{\circ}) + sin 360^{\circ}$

Now, we know $sin (π + θ) = - sin θ$

$\Rightarrow sin (180^{\circ} + 10^{\circ}) + sin (180^{\circ} + 20^{\circ}) + sin (180^{\circ} + 30^{\circ}) + \dots + sin (180^{\circ} + 170^{\circ}) + sin 360^{\circ} = - sin 10^{\circ} - sin 20^{\circ} - sin 30^{\circ} - \dots - sin 170^{\circ} + 0$
Now, adding these with thw first $18$ terms, we get:
$sin 10^{\circ} + sin 20^{\circ} + sin 30^{\circ} + \dots + sin 170^{\circ} + sin 180^{\circ} + sin 190^{\circ} sin 200^{\circ} + \dots + sin 350^{\circ} + sin 360^{\circ} = sin 10^{\circ} + sin 20^{\circ} + sin 30^{\circ} + \dots + sin 170^{\circ} + 0 - sin 10^{\circ} - sin 20^{\circ} - \dots - sin 170^{\circ} = 0$

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Similar questions

sin 10^{\circ} + sin 20^{\circ} + sin 30^{\circ} + . . . . + sin 360^{\circ}

sin10°+sin20°+sin30°+....+sin360°=?

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s i n 10^{0} + s i n 20^{0} + s i n 30^{0} + . . . . + s i n 360^{0}

sin 10^{\circ} + sin 20^{\circ} + sin 40^{\circ} + sin 5 θ^{\circ} = sin 70^{\circ} + sin 80^{\circ}

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