Without using tables find the value of tan 75 c... - SS3 Mathematics Trigonometric Identities and Equations Question
Without using tables, find the value of \(\tan{75{^\circ}}\) in surd form
\(\frac{2 + \sqrt{3}}{2 - \sqrt{3}}\)
\(\frac{2 - \sqrt{3}}{2 - \sqrt{3}}\)
\(\frac{2}{2 + \sqrt{3}}\)
\(\frac{2}{\sqrt{3}}\)
\[\tan{75{^\circ}} = \tan{(45{^\circ} + 30{^\circ})}\]
\[\tan{(A + B)} = \frac{\sin(A + B)}{\cos(A + B)} = \frac{\tan A + \tan B}{1 - \tan A\tan B}\]
\[\tan{(45{^\circ} + 30{^\circ})} = \frac{\sin(45{^\circ} + 30{^\circ})}{\cos(45{^\circ} + 30{^\circ})} = \frac{\tan{45{^\circ}} + \tan{30{^\circ}}}{1 - \tan{45{^\circ}}\tan{30{^\circ}}}\]
\[\tan{(45{^\circ} + 30{^\circ})} = \frac{\tan{45{^\circ}} + \tan{30{^\circ}}}{1 - \tan{45{^\circ}}\tan{30{^\circ}}} = \frac{1 + \frac{\sqrt{3}}{2}}{1 - (1)(\frac{\sqrt{3}}{2})} = \frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{\frac{2}{2} - \frac{\sqrt{3}}{2}}\]
\(\frac{\frac{2 + \sqrt{3}}{2}}{\frac{2 - \sqrt{3}}{2}} = \frac{2 + \sqrt{3}}{2} \times \frac{2}{2 - \sqrt{3}} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}\)
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