What is the value of $sin 10^{o} + sin 20^{o} + sin 40^{o} + sin 50^{o}$ ?

sin 70^{o} - sin 80^{o}

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sin 60^{o} + sin 80^{o}

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sin 40^{o} - sin 80^{o}

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sin 70^{o} + sin 80^{o}

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Solution

The correct option is B $sin 70^{o} + sin 80^{o}$
$sin 10^{\circ} + sin 20^{\circ} + sin 40^{\circ} + sin 50^{\circ}$
$\to (sin 50^{\circ} + sin 10^{\circ}) + (sin 40^{\circ} + sin 20^{\circ})$ : $(sin A + sin B = 2 sin \frac{(A + B)}{2} cos \frac{(A - B)}{2})$
$\to (2 sin 30^{\circ} cos 20^{\circ}) + (2 sin 30^{\circ} cos 10^{\circ})$ : $(sin 30^{\circ} = \frac{1}{2})$
$\to (2 \times \frac{1}{2} \times cos 20^{\circ}) + (2 \times \frac{1}{2} \times cos 10^{\circ})$

$\to cos 20^{\circ} + cos 10^{\circ}$
$\to sin (90^{\circ} - 20^{\circ}) + sin (90^{\circ} - 10^{\circ})$ $(sin (90^{\circ} - θ) = cos θ)$
$\to sin 70^{\circ} + sin 80^{\circ}$

so the correct option will be $D$

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Similar questions

\frac{sin 10^{o} + sin 50^{o} - sin 80^{o} + sin 140^{o}}{\sqrt{2} . sin 25^{o}}

\frac{cos 10^{o}}{sin 80^{o}} + sin 42^{o} \times 4 sec 48^{o}

\frac{sin 70^{o} + cos 40^{o}}{cos 70^{o} + sin 40^{o}}

\frac{sin 70^{o} + cos 40^{o}}{cos 70^{o} + sin 40^{o}} =

\frac{sin 70^{o}}{cos 20^{o}} + \frac{cosec 20^{o}}{sec 70^{o}} - 2 cos 70^{o} cosec 20^{o}

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