Evaluate \(\cos 75°\), leaving the answer in surd form.
FURTHER MATHEMATICS
WAEC 2016
Evaluate \(\cos 75°\), leaving the answer in surd form.
- A. \(\frac{\sqrt{2}}{2}(\sqrt{3} + 1)\)
- B. \(\frac{\sqrt{2}}{4}(\sqrt{3} - 1)\)
- C. \(\frac{\sqrt{2}}{4}(\sqrt{3} + 1)\)
- D. \(\frac{\sqrt{2}}{2}(\sqrt{3} - 1)\)
Correct Answer: B. \(\frac{\sqrt{2}}{4}(\sqrt{3} - 1)\)
Explanation
\(\cos(a + b) = \cos a\cos b - \sin a\sin b\)
\(\cos75° = \cos(30 + 45) = (\cos30)(\cos45) - (\sin30)(\sin45)\)
= \((\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}) - (\frac{1}{2} \times \frac{\sqrt{2}}{2})\)
= \(\frac{\sqrt{6} - \sqrt{2}}{4}\)
= \(\frac{\sqrt{2}(\sqrt{3} - 1)}{4}\)
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