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WAEC Further Mathematics 2025 Paper

Question : 33

Total: 40

Find in surd form, the value of cos165°.

Solution:

Concept:

Use the cosine compound-angle identities and quadrant signs to express

\cos 165^\circ

in surd form.

Explanation:

Write

165^\circ

180^\circ - 15^\circ

Since

\cos(180^\circ - \theta) = -\cos\theta

, we get

\cos165^\circ = -\cos15^\circ

Now calculate

\cos15^\circ = \cos(45^\circ - 30^\circ)

Using

\cos(A-B) = \cos A\cos B + \sin A\sin B

, we have:

\cos15^\circ = \cos45^\circ\cos30^\circ + \sin45^\circ\sin30^\circ

Substitute the standard values:

\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2}

This equals

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

Therefore,

\cos165^\circ = -\frac{\sqrt{6}+\sqrt{2}}{4} = -\frac{1}{4}(\sqrt{6}+\sqrt{2})

Answer:

-\frac{1}{4}(\sqrt{6}+\sqrt{2})

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