Question

Need help with a complex numbers problem. Write -7+4i in trigonometric form.
I know that R = sqrt 65 and theta = 0.519 ( Due to Arctan -4/7) but I don't know how to get the values of sine and cosine.

Answer 1

you don't get the values of sine and cosine, you have them, they are \(-7\) for cosine and \(4\) for sine
what you need is \(r\) and \(\theta\)

Answer 2

unfortunately \(\theta\) is not \(\tan^{-1}(-\frac{4}{7})\) because you are in quadrant II

Answer 3

\[-7+4i=r\left(\cos(\theta)+i\sin(\theta)\right)\]

Answer 4

\[r\cos(\theta)=-7, r\sin(\theta)=4\]
i made a mistake when i said \(\sin(\theta)=4\) etc, of course that is not possible.
what i meant to say was \(r\sin(\theta)=4\)

Answer 5

your \(r\) is correct, it is \(\sqrt{7^2+4^2}=\sqrt{65}\) but your \(\theta\) is wrong

Answer 6

How would you get theta in this situation? I got Tan(theta) = b/a = 4/-7 then I used arc tan to get my answer which gives me -.519146

Answer 7

are you working in degrees or radians?

Answer 8

Radians

Answer 9

you can only use \(\theta=\tan^{-1}(\frac{b}{a})\) if you are in quadrant I or IV because that is the range of arctangent, but you are in quadrant 2

Answer 10

\(\theta\) is not unique, you can go around the circle again and again to come to a coterminal angle, but the one you found, \(-.519146\) is in the wrong place, it is in quadrant 4

Answer 11

easy enough to fix though, add \(\pi\) to your answer

Answer 12

is this more or less clear? arctangent only returns answers in the interval \([-\frac{\pi}{2},\frac{\pi}{2}]\) and you are not in that interval