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WAEC Further Mathematics 2025 Paper

Question : 33

Total: 40

Find in surd form, the value of cos165°.

Solution:

To find

\cos 165^\circ

, we use the cosine of the difference formula:

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

Expressing

165^\circ

180^\circ - 15^\circ

\cos 165^\circ = \cos(180^\circ - 15^\circ) = -\cos 15^\circ

Now, finding

\cos 15^\circ

using

45^\circ - 30^\circ

\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ

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

\cos 30^\circ = \frac{\sqrt{3}}{2}

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

\sin 30^\circ = \frac{1}{2}

\cos 15^\circ = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)

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

Thus,

\cos 165^\circ = -\cos 15^\circ = -\frac{\sqrt{6} + \sqrt{2}}{4}

= -

\frac{1}{4}

(

\sqrt{6}

\sqrt{2}

)

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