Find the value of (60^{} + 45^{}) leaving your answer in surd form.

Correct Answer is : 2−64{{2} - {6}}{4}42−6

Correct Answer is : 2−64{{2} - {6}}{4}42−6

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WAEC Further Mathematics 2010 Paper

Question : 2

Total: 40

\cos(60^{\circ} + 45^{\circ})

leaving your answer in surd form.

Solution:

Concept:

Use the compound angle formula for cosine to expand

\cos(60^{\circ} + 45^{\circ})

Explanation:

The formula is

\cos(x + y) = \cos x \cos y - \sin x \sin y

Here,

x = 60^{\circ}

and

y = 45^{\circ}

Substitute the standard values:

\cos 60^{\circ} = \frac{1}{2}

\cos 45^{\circ} = \frac{\sqrt{2}}{2}

\sin 60^{\circ} = \frac{\sqrt{3}}{2}

, and

\sin 45^{\circ} = \frac{\sqrt{2}}{2}

Thus,

\cos(60^{\circ} + 45^{\circ}) = \frac{1}{2} \times \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}

Simplify each product:

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

Combine the numerators over the common denominator:

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

Answer:

\cos(60^{\circ} + 45^{\circ}) = \frac{\sqrt{2} - \sqrt{6}}{4}

, which corresponds to option C.

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