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:

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

In this case, we are given:

$z=-8+2i$

We need to find r and θ. We know:

· a = -8

· b = 2

Thus:

$r=\sqrt{(-8)^2+2^2}=\sqrt{64+4}=\sqrt{68}=2\sqrt{17}$

And now, we can find θ:

$-8=2\sqrt{17}\cos\theta$

And solve:

$\begin{gathered} \cos\theta=-\frac{8}{2\sqrt{17}} \\ . \\ \theta=\cos^{-1}(-\frac{4\sqrt{17}}{17})\approx2.89661 \end{gathered}$

Now we can write the number in trigonometric form:

$z=2\sqrt{17}(\cos(2.9)+i\sin(2.9))$