Without using tables find the value of cos 120 ... - SS3 Mathematics Trigonometric Identities and Equations Question
Without using tables, find the value of \(\cos{(120{^\circ} + 45{^\circ}})\) in surd form.
\[\cos(A + B) = \cos A\cos B - \sin A\sin B\]
\[\cos(120{^\circ} + 45{^\circ}) = \cos{120{^\circ}}\cos{45{^\circ}} - \sin{120{^\circ}}\sin{45{^\circ}}\]
Since \(\cos{120{^\circ}} = - \cos{60{^\circ}}\) and \(\sin{120{^\circ}} = \sin 60{^\circ}\)
\[\cos(120{^\circ} + 45{^\circ}) = - \cos{60{^\circ}}\cos{45{^\circ}} - \sin{60{^\circ}}\sin{45{^\circ}}\]
\(\cos(120{^\circ} + 45{^\circ}) = - \left( \frac{1}{2} \right)\left( \frac{\sqrt{2}}{2} \right) - \left( \frac{\sqrt{3}}{2} \right)\left( \frac{\sqrt{2}}{2} \right)\)
\(= - \frac{\sqrt{2}}{4} - \frac{\sqrt{2}.\sqrt{3}}{4} = \frac{- \sqrt{2} - \sqrt{2}.\sqrt{3}}{4} = \frac{- \sqrt{2}(1 + \sqrt{3})}{4}\)
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