First identify which quadrant we are in:
x = real = -12
y = imaginary = 16
(-12,16) is in 2nd quadrant, this means theta is between 90 and 180.
Next calculate "r", which is distance from (-12,16) to origin:
\(r = \sqrt{x^2 +y^2} = \sqrt{12^2 + 16^2} = \sqrt{400} = 20\)
Finally, calculate theta:
\(\theta = \tan^{-1}(\frac{y}{x}) = \tan^{-1} (\frac{16}{-12}) = 126.87\)
Note: when you put this in your calculator it will give you -53.13 (4th quadrant)
Just add 180 so that angle is in correct quadrant.
Final Answer:
\(-12 + 16i = 20(\cos 126.87 + i \sin 126.87)\)
