Question

Write the trigonometric form of the complex number. (Round your angles to two decimal places. Let 0 ≤ < 2.)

Answer 1

Given a complex number z:

`z=a+bi`

We can write this number in trigonometric form, using:

`\begin{gathered} z=r(\cos\theta+i\sin\theta) \\ . \\ a=r\cos\theta \\ . \\ b=r\sin\theta \\ . \\ r=√(a^2+b^2) \end{gathered}`

In this case, we are given:

`z=-8+2i`

We need to find r and θ. We know:

· a = -8

· b = 2

Thus:

`r=√((-8)^2+2^2)=√(64+4)=√(68)=2√(17)`

And now, we can find θ:

`-8=2√(17)\cos\theta`

And solve:

`\begin{gathered} \cos\theta=-(8)/(2√(17)) \\ . \\ \theta=\cos^(-1)(-(4√(17))/(17))\approx2.89661 \end{gathered}`

Now we can write the number in trigonometric form:

`z=2√(17)(\cos(2.9)+i\sin(2.9))`