# 3blue1brown's complex numbers fundamental video: rotating a point by 30 degrees

Grant says that multiplying any complex number represented by a point on the Argand plane by $$cis(30^{\circ})$$
i.e. $$\cos30^{\circ} +i\sin(30^{\circ})$$ rotates that point by $$30$$ degrees counterclockwise on the Argand plane. Maybe I did not pick on what he was trying to say in the video, but can someone elaborate on why this happens? Even people who haven't watched the video........

Here's a link by the way: https://www.youtube.com/watch?v=5PcpBw5Hbwo . Start watching after 32:10. I get why multiplying $$cis(30^{\circ})$$ to $$1$$, rotates $$1$$ by $$30$$ degrees counterclockwise, but can someone algebraically prove it for any general complex number (z)? It's easy to see with the Euler notation... but I am not well-worsed with it ( only know the basics ). I am requesting a proof strictly in $$a + ib$$ form and then elaborating on that geometrically to show why a point might rotate by $$30$$ degrees. I am sorry if I'm not able to articulate exactly what I mean, I'd be more than happy to clarify in the comments section.

Let $$z = re^{i\theta}$$ be a complex number where $$r\gt 0$$ and $$-\pi\lt \theta \le \pi$$. Consider another complex number which we want to multiply, $$z_1 = r_1e^{i\theta_1}$$ where $$r_1\gt 0$$ and $$-\pi\lt \theta_1 \le \pi$$. Then we have: $$z_p =zz_1 = (rr_1)(e^{i(\theta + \theta_1)})$$
So the product is another complex number and $$|z_p| = rr_1 = |z||z_1|$$. Also $$Arg(z_p) = Arg(z) + Arg(z_1) +2k\pi$$ and $$k$$ can be $$0, 1$$ or $$-1$$. For example if $$\theta_1 = \frac{\pi}{6}$$, then: $$-\pi\lt\theta \le \frac{5\pi}{6} \implies -\frac{5\pi}{6}\lt\theta + \theta_1\le \pi \implies Arg(z_p) = \theta + \theta_1$$ $$\frac{5\pi}{6}\lt\theta \le \pi \implies \pi \lt\theta + \theta_1\le \frac{7\pi}{6} \implies Arg(z_p) = \theta + \theta_1 - 2\pi$$