Find, in surd form, the value of \(\cos 165\).
FURTHER MATHEMATICS
WAEC 2006
Find, in surd form, the value of \(\cos 165\).
- A. \(\frac{1}{4}(\sqrt{6} + \sqrt{2})\)
- B. \(\frac{1}{4}(\sqrt{6} - \sqrt{2})\)
- C. \(-\frac{1}{4}(\sqrt{6} - \sqrt{2})\)
- D. \(-\frac{1}{4}(\sqrt{6} + \sqrt{2})\)
Correct Answer: D. \(-\frac{1}{4}(\sqrt{6} + \sqrt{2})\)
Explanation
\(\cos 165 = -\cos (180 - 165) = -\cos 15\)
\(\cos 15 = \cos (45 - 30)\)
\(\cos (x - y) = \cos x \cos y + \sin x \sin y\)
\(\cos (45 - 30) = \cos 45 \cos 30 + \sin 45 \sin 30\)
= \((\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})\)
= \(\frac{1}{4}(\sqrt{6} + \sqrt{2})\)
\(\therefore \cos 165 = -\frac{1}{4}(\sqrt{6} + \sqrt{2})\)
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