MCQ
Sum
cos 35° + cos 85° + cos 155° =
Options
0
- \[\frac{1}{\sqrt{3}}\]
- \[\frac{1}{\sqrt{2}}\]
cos 275°
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Solution
0
\[\cos35^\circ + \cos85^\circ + \cos155^\circ\]
\[ = 2\cos\left( \frac{35^\circ + 85^\circ}{2} \right) \cos\left( \frac{35^\circ - 85^\circ}{2} \right) + \cos155^\circ \left[ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = 2\cos60^\circ \cos\left( - 25^\circ \right) + \cos155^\circ\]
\[ = 2 \times \frac{1}{2}\cos25^\circ + \cos155^\circ\]
\[ = \cos25^\circ + \cos155^\circ\]
\[ = 2\cos\left( \frac{25^\circ + 155^\circ}{2} \right) \cos\left( \frac{25^\circ - 155^\circ}{2} \right)\]
\[ = 2\cos90^\circ \cos65^\circ\]
\[ = 0\]
\[ = 2\cos\left( \frac{35^\circ + 85^\circ}{2} \right) \cos\left( \frac{35^\circ - 85^\circ}{2} \right) + \cos155^\circ \left[ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = 2\cos60^\circ \cos\left( - 25^\circ \right) + \cos155^\circ\]
\[ = 2 \times \frac{1}{2}\cos25^\circ + \cos155^\circ\]
\[ = \cos25^\circ + \cos155^\circ\]
\[ = 2\cos\left( \frac{25^\circ + 155^\circ}{2} \right) \cos\left( \frac{25^\circ - 155^\circ}{2} \right)\]
\[ = 2\cos90^\circ \cos65^\circ\]
\[ = 0\]
Concept: Transformation Formulae
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