Question

Write the complex number in trigonometric form express the angle and radiance do not round any intermediate computations round the values in your answer to the nearest hundredth

Answer 1

Given a complex number in trigonometric form below:

$\begin{gathered} x+yi=r(\cos \theta+i\sin \theta),where \\ r=\sqrt[]{x^2+y^2} \\ \theta=\tan ^{-1}(\frac{y}{x}),-\pi<\theta\leq\pi \end{gathered}$

Given

$z=6+4i$$\begin{gathered} x=6 \\ y=4 \\ r=\sqrt[]{6^2+4^2} \\ r=\sqrt[]{36+16} \\ r=\sqrt[]{52} \end{gathered}$$\begin{gathered} \theta=\tan ^{-1}(\frac{4}{6}) \\ \theta=\tan ^{-1}(0.6667) \\ \theta=33.69^0 \end{gathered}$

Convert 33.69 degrees to radian

$33.69^0=0.588rad$

Since tan is positive in the first and third quadrants, the value of teetha would be

$\begin{gathered} 1st\text{ quadrant,} \\ \theta=33.69^0=0.588rad \\ 4th\text{ quadrant} \\ \theta=180^0+33.69^0 \\ \theta=213.69^0=3.73\text{rad} \end{gathered}$

Therefore,

$\begin{gathered} z=6+4i \\ z=\sqrt[]{52}(\cos 0.588+i\sin 0.588),or \\ z=\sqrt[]{52}(\cos 3.73+i\sin 3.73) \end{gathered}$

Hence, the trigonometric form 6+4i is √52(cos0.588+isin0.588)