Question

Which of the following represents z equals negative 5 radical 3 end radical minus 5 times i in trigonometric form?

Answer 1

SOLUTIONS

This is the trigonometric form of a complex number where

$z=-5\sqrt{3}-5i$

the modulus and

θ

is the angle created on the complex plane.

From the graph, a = r cos θ and b = r sin θ.

z=a+bi

z=rcosθ+irsinθ

z=r(cosθ+isinθ)

$\begin{gathered} r=\sqrt{a^2+b^2} \\ r=\sqrt{5\sqrt{3})^2+5^2} \\ r=\sqrt{75+25} \\ r=\sqrt{100} \\ r=10 \end{gathered}$

Trigonometric Form of a Complex Number

z=r(cosθ+isinθ)

r is called the modulus and θ is called the argument

Convert between trigonometric form and standard form using

a=rcosθ

b=rsinθ

tanθ=b/a

$\begin{gathered} tan\theta=\frac{b}{a}=\frac{-5}{-5\sqrt{3}}=\frac{1}{\sqrt{3}} \\ \theta=tan^{-1}(\frac{1}{\sqrt{3}}) \\ \theta=210 \end{gathered}$

Therefore the trigonometric form will be