Question

Find this solution for mathematical quiz

Answer 1

Answer:

$-\sqrt{2} + \sqrt{2}i$

Step-by-step explanation:

Angle of 9pi/4

The equivalent angle of $\frac{9\pi}{4}$, on the first lap, is found subtracting this angle from $2\pi$. Thus:

$\frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}$

Thus, the sine and cosine are:

$\sin{(\frac{9\pi}{4})} = \sin{(\frac{\pi}{4})} = \frac{\sqrt{2}}{2}$

$\cos{(\frac{9\pi}{4})} = \cos{(\frac{\pi}{4})} = \frac{\sqrt{2}}{2}$

Angle of 3pi/2

On the first lap of the circle, thus no need to find the equivalent angle. We have that:

$\sin{(\frac{3\pi}{2})} = -1, \cos{(\frac{3\pi}{2})} = 0$

Expression:

$4(\cos{(\frac{9\pi}{4})} + i\sin{(\frac{9\pi}{4})}) \div 2(\cos{(\frac{3\pi}{2})} + i\sin{(\frac{3\pi}{2})})$

$4(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}) \div 2(0 - i)$

$\frac{2\sqrt{2} + 2\sqrt{2}i}{-2i} \times \frac{i}{i}$

Considering that $i^2 = -1$

$\frac{-2\sqrt{2}+2\sqrt{2}i}{2}$

$-\sqrt{2} + \sqrt{2}i$