# Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.2 Trigonometric (Polar) Form of Complex Numbers - 8.2 Exercises - Page 371: 52

$2(\cos(270^{\circ})+i\sin(270^{\circ}))$

#### Work Step by Step

To find the trigonometric form of a complex number from its Cartesian form, two things must be done. Firstly, the modulus of the complex number must be found. Secondly, the argument of the complex number must be found. To find the modulus of the complex number: Apply Pythagoras theorem to the coefficients of the complex number. i.e. $r=\sqrt{(-2)^{2}}\\=2$ To find the argument of the complex number: First, find the basic argument of the complex number this is done by finding $\arctan$ of the fraction of the absolute value of the imaginary number coefficient over the coefficient of the real number. . The coefficient of the imaginary number, being negative, suggest that the argument is $270^{\circ}$ as measured anticlockwise from the positive real axis to the vertical line of the Cartesian equation. Thus, the trigonometric form of the equation of the Cartesian complex number $-2i$ is $2(\cos(270^{\circ})+i\sin(270^{\circ}))$

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