Write the complex number in trigonometric form, once using degrees and once using radians. Begin by sketching the graph to help find the argument θ. (Do not use cis form.)
$$−1 + i$$
My work:
I graphed $x = -1$ and $y = 1$
$$z=r= \sqrt{ x^2 + y^2}$$
$$r= \sqrt{2}$$
$$\tan \theta = \frac{Opposite}{Adjacent} $$
$$\tan \theta = \frac{-1}{1} = -1$$
$$\theta= 45^\circ$$
When put into trig form: $$\sqrt{2} (\cos 45^\circ +i \sin 45^\circ)$$
Here is how my submitted answer looks (it is #9):
I also need help with $9 − 40i$ (instructions: convert the complex number to trigonometric form. (Enter the angle in degrees rounded to two decimal places. Do not use cis form.).
I went through the same steps as I did on the other problem, and I got $r=41$ and $θ= -77.32$.
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$\begingroup$You have that:$x = - 1, y = 1, z = x+iy \Rightarrow \theta = \pi+\tan^{-1}\left(\dfrac{y}{x}\right) = \pi+\tan^{-1}(-1) = \pi+\dfrac{-\pi}{4}=\dfrac{3\pi}{4}, r= \sqrt{x^2+y^2}=\sqrt{2}$
$\endgroup$ $\begingroup$You should bear in mind that an arctan will always output an answer between -90 and 90, so you may have to add or subtract 180 from the answer to get the appropriate angle, because if you examine your graph, the number appears not to be at -45.
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