Evaluate 75^{}, leaving the answer in surd form.

Correct Answer is : 24(3−1){{2}}{4}({3} - 1)42(3−1)

Correct Answer is : 24(3−1){{2}}{4}({3} - 1)42(3−1)

Test Index

WAEC Further Mathematics 2016 Paper

Question : 21

Total: 40

Evaluate

\cos 75^{\circ}

, leaving the answer in surd form.

Solution:

Concept:

Use the compound angle formula for cosine.

Explanation:

Write

75^{\circ}

45^{\circ} + 30^{\circ}

Then

\cos 75^{\circ} = \cos(45^{\circ} + 30^{\circ})

Using

\cos(a+b) = \cos a \cos b - \sin a \sin b

, we get:

\cos 75^{\circ} = \cos 45^{\circ}\cos 30^{\circ} - \sin 45^{\circ}\sin 30^{\circ}

Substitute the standard values:

= \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)

= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}

= \frac{\sqrt{6} - \sqrt{2}}{4}

Factor out

\sqrt{2}

= \frac{\sqrt{2}(\sqrt{3} - 1)}{4}

Answer:

\frac{\sqrt{2}}{4}(\sqrt{3} - 1)

, which is Option B.

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