FP2 January 2006 Q5
5. Solve the equation \[z^5 = \mathrm{i}\] giving your answers in the form \(\cos\theta + \mathrm{i}\sin\theta\).
| Scheme | Marks |
|---|---|
| \(\cos\dfrac{\pi}{2} + \mathrm{i}\sin\dfrac{\pi}{2}\) | B1 |
| \(\cos\dfrac{\pi}{10} + \mathrm{i}\sin\dfrac{\pi}{10}\) | B1 |
| \(\cos\left(\dfrac{(4k + 1)\pi}{10}\right) + \mathrm{i}\sin\left(\dfrac{(4k + 1)\pi}{10}\right),\ k = 2, 3, 4\) (or equiv.) \(\left[\cos\left(\dfrac{9\pi}{10}\right) + \mathrm{i}\sin\left(\dfrac{9\pi}{10}\right),\ \cos\left(\dfrac{13\pi}{10}\right) + \mathrm{i}\sin\left(\dfrac{13\pi}{10}\right),\ \cos\left(\dfrac{17\pi}{10}\right) + \mathrm{i}\sin\left(\dfrac{17\pi}{10}\right)\right]\) [Degrees: 18, 90, 162, 234, 306] | M1 A2, 1, 0 |
| (5 marks) |