Multiple Angles - SS3 Mathematics Lesson Note

1. \(\sin{2A} = 2\sin A\cos A\)

2. \(\cos{2A} = 1 - 2\sin^{2}A\) or \(\cos{2A} = 2\cos^{2}A - 1\)

3. \(\tan{2A} = \ \frac{2\tan A}{1 - \tan^{2}A}\)

4. \(\sin{3A} = \ 3\sin A - 4\sin^{3}A\)

5. \(\cos{3A} = 4\cos^{3}A - 3\cos A\)

6. \(\tan{3A} = \ \frac{3\tan A - \tan^{3}A}{1 - 3\tan^{3}A}\)

7. \(\sin^{2}A = \frac{1}{2}(1 - \cos{2A})\)

8. \(\cos^{2}A = \frac{1}{2}(1 + \cos{2A})\)

Example 4 Evaluate \(\tan{2\theta}\), if \(\tan\theta = \frac{4}{3}\)

Solution

\[\tan{2\theta} = \ \frac{2\tan\theta}{1 - \tan^{2}\theta}\]

\[= \frac{2(\frac{4}{3})}{1 - {(\frac{4}{3})}^{2}} = \ \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{- \frac{7}{9}} = \frac{8}{3} \times \frac{- 9}{7} = - \frac{24}{7}\]