Question

Which of the following represents z equals negative 4 radical 3 end radical plus 4 times i in trigonometric form?

Answer 1

Notice that

`\begin{gathered} z=-4\sqrt[]{3}+4i=8(-\frac{\sqrt[]{3}}{2}+(1)/(2)i) \\ \Rightarrow z=8(-\frac{\sqrt[]{3}}{2}+(1)/(2)i) \end{gathered}`

Set,

`\begin{gathered} z=8(\cos x+i\sin x) \\ \Rightarrow8(\cos x+i\sin x)=8(-\frac{\sqrt[]{3}}{2}+(1)/(2)i) \\ \Rightarrow\cos x=-\frac{\sqrt[]{3}}{2},\sin x=(1)/(2) \end{gathered}`

Then, angle x is in the second quadrant since cosx is negative and sinx is positive-

On the other hand,

Therefore, using the unitary circle,

Therefore, the angle that produces such values for the cosine and sine function is x=150°; thus,

`\begin{gathered} x=150\degree \\ \Rightarrow z=8(-\frac{\sqrt[]{3}}{2}+(1)/(2)i)=8(\cos (150\degree+i\sin (150\degree))) \\ \Rightarrow z=8(\cos (150\degree+i\sin (150\degree))) \end{gathered}`

The answer is option 4 (top to bottom)