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Question
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
Options
- \[\frac{1}{2}\]
- \[- \frac{1}{2}\]
−1
None of these
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Solution
None of these
Explanation:
= \[\sin78^\circ - \sin66^\circ - \sin42^\circ + \sin60^\circ\]
\[ = \sin78^\circ - \sin42^\circ - \sin66^\circ + \sin60^\circ\]
\[ = 2\sin\left( \frac{78^\circ - 42^\circ}{2} \right)\cos\left( \frac{78^\circ + 42}{2} \right) - \sin66^\circ + \sin60^\circ \left[ \because \sin A - \sin B = 2\sin\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right) \right]\]
\[ = 2\sin18^\circ \cos60^\circ - \sin66^\circ + \sin60^\circ\]
\[ = 2 \times \frac{1}{2}\sin18^\circ - \sin66^\circ + \frac{\sqrt{3}}{2}\]
\[ = \sin18^\circ - \sin66^\circ + \frac{\sqrt{3}}{2}\]
\[ = \frac{\sqrt{5} - 1}{4} - 0 . 914 + \frac{\sqrt{3}}{2}\]
= 0.309 − 0.914 + 0.866
= 0.261
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