Respuesta :
Question: Express [tex]4i[/tex] in trigonometric form.
Answer:
A few ways to express this is:
[tex]4(\cos(\frac{\pi}{2})+\sin(\frac{\pi}{2})i)[/tex]
[tex]4cis(\frac{\pi}{2})[/tex]
[tex]4e^{i\frac{\pi}{2}}[/tex]
Choose the notation your class is using.
Step-by-step explanation:
[tex]r(\cos(\theta)+\sin(\theta)i)[/tex] is trigonometric form.
Distributing [tex]r[/tex] to terms inside the ( ) gives:
[tex]r\cos(\theta)+r\sin(\theta)i[/tex].
Our number has no real part so we are just trying to find [tex]r[/tex] and [tex]\theta[/tex] such that [tex]r\sin(\theta)=4[/tex].
Since we want [tex]\cos(\theta)=0[/tex], we will have to [tex]\sin(\theta)=1[/tex] (or -1).
This happens at [tex]\frac{\pi}{2}[/tex].
This means [tex]\theta=\frac{\pi}{2}[/tex].
So while [tex]\sin(\theta)=1[/tex] this forces [tex]r=4[/tex]
The trigonometric form is [tex]4(\cos(\frac{\pi}{2})+\sin(\frac{\pi}{2})i)[/tex].
Some people like to express this as [tex]4cis(\frac{\pi}{2})[/tex] which is way shorter.
Some people might also express this as [tex]4e^{i\frac{\pi}{2}}[/tex].
