Which of the following statements about complex numbers is true?
- For all real numbers \(x, y, \theta\) with \(x \neq 0\),
\(\tan \theta=\dfrac{y}{x} \ \Rightarrow \ x+i y=r e^{i \theta}\), for some real number \(r\).
- For all non-zero complex numbers \(z_1\) and \(z_2\),
\(\operatorname{Arg}\left(z_1\right)=\theta_1\) and \(\operatorname{Arg}\left(z_2\right)=\theta_2 \ \Rightarrow \ \operatorname{Arg}\left(z_1 z_2\right)=\theta_1+\theta_2,\)
where \(\operatorname{Arg}\) denotes the principal argument.
- For all real numbers \(r_1, r_2, \theta_1, \theta_2\) with \(r_1, r_2>0\),
\(r_1 e^{i \theta_1}=r_2 e^{i \theta_2} \ \Rightarrow \ r_1=r_2\) and \(\theta_1=\theta_2 \text {. }\)
- For all real numbers \(x, y, r, \theta\) with \(r>0\) and \(x \neq 0\),
\(x+i y=r e^{i \theta} \ \Rightarrow \ \theta=\arctan \Big(\dfrac{y}{x} \Big)\)