Question12. Find the square root of (a) $5+12 i$

Studdy Solution
Use the modulus and argument to find the square root of the complex number. The square root is given by:
$$\sqrt{r}(\cos(\theta/2) + i\sin(\theta/2))$$
Calculate $\sqrt{r}$:
$$\sqrt{r} = \sqrt{13}$$
Calculate $\theta/2$:
$$\theta/2 \approx 0.588 \text{ radians}$$
Calculate $\cos(\theta/2)$ and $\sin(\theta/2)$:
$$\cos(0.588) \approx 0.832$$ $$\sin(0.588) \approx 0.555$$
Thus, the square root of $5 + 12i$ is:
$$\sqrt{13}(0.832 + 0.555i)$$
Calculate the final expression:
$$\approx 3.0 + 2.0i$$
The square root of $5 + 12i$ is approximately:
$$\boxed{3.0 + 2.0i}$$