# Please explain why z = cosθ + jsinθ in this video.

• anhnha
Hi, I meant why z = cosθ + jsinθ not Euler's formula.
Is z a vector? Why not use vector symbol anywhere here? I can't figure out the equation z = cosθ + jsinθ.

Hi, I meant why z = cosθ + jsinθ not Euler's formula.
Is z a vector? Why not use vector symbol anywhere here? I can't figure out the equation z = cosθ + jsinθ.

Euler's formula. The proof is in the link I posted. We can write $z = e^{i\theta}$ (since z lies on the unit circle), and the rest follows immediately from Euler.

How much experience do you have with complex variables? I'm really not sure where your confusion lies. The video explains the equation is a fair bit of detail, and the link I posted gives a rigorous proof, so if you're still confused you're going to have to be more specific.

Hi again,
I use complex number a lot but not really understand it.
My confusion is in the equation z = cosθ + j sinθ
z is the distance from origin O to z point. It is a real number
But cosθ + j sinθ is clearly a complex number.
A real number is impossible to equal to a complex number. That is what I can't get.

Hi again,
I use complex number a lot but not really understand it.
My confusion is in the equation z = cosθ + j sinθ
z is the distance from origin O to z point. It is a real number
But cosθ + j sinθ is clearly a complex number.
A real number is impossible to equal to a complex number. That is what I can't get.

z is not "the distance from the origin O to z point", it's a complex number. This is stated very explicitly in the video.

z = cosθ + jsinθ is a unit circle in the complex plane; you could also write it as r=(cosθ,sinθ) ... then when you check the Euclidean distance you will find r^2 = cos^2 θ + sin^2 θ ... which gives r^2 = 1.

Complex numbers have magnitudes; it is their magnitude |z| which gives the distance to the origin. In this case we have |z|=1. The complex number z also represents a direction in the complex plane; here it is clear that the direction is given by the angle which rotates about the origin.

Thanks. I have misunderstood it for a long time. :(