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)((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)((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)