Euler's Formula: the Complex Exponential on the Imaginary Axis, Its Modulus and Its Conjugate

In the setting of Complex Analysis: Standing Notation, let $\exp:\mathbb{C}\to\mathbb{C}$ be the complex exponential function, let $\cos,\sin:\mathbb{R}\to\mathbb{R}$ be the cosine and sine functions, and let $t,x,y\in\mathbb{R}$.

(i)Euler's formula.

$$\exp(it)=\cos t+i\sin t .$$

(ii)Unit modulus. $|\exp(it)|=1$.

(iii)The exponential in real coordinates.

$$\exp(x+iy)=\exp_{\mathbb{R}}(x)\,(\cos y+i\sin y).$$

(iv)Conjugates.

$$\overline{\exp(it)}=\exp(-it).$$