First, divide the expression so that you can get a complex number in a+bi form. To do this, you must multiply the numerator and denominator by the conjugate, i-4.
(-i-5)/(i+4)
=(-i-5)/(i+4) * (i-4)/(i-4) ->multiply by the conjugate
=(-i^2 + 4i - 5i +20)/(i^2-4^2) ->expand
=(1 + 4i - 5i +20)/(-1-16) ->simplify
=(21 -i)/(-17) ->combine like terms
=color(blue)(-21/17 + 1/17i) ->rewrite in a+bi form
To convert this to trigonometric form, you must find out r, the distance from the origin to the point, and theta, the angle. Use the following formulas:
r=sqrt(a^2+b^2)
tan theta = b/a
In this case, a=-21/17 and b=1/17.
r=sqrt(a^2+b^2) = sqrt((-21/17)^2+(1/17)^2) ~~ color(blue)1.24
tan theta = (1/17)/(-21/17) = 1/17 * -17/21 = -1/21
theta=tan^-1(-1/21) ~~ -0.05
However, since the coordinate (-21/17 + 1/17i) is in Quadrant II, this angle is wrong. This is because we used the arctan function, which doesn't account for angles outside of the range [-pi/2, pi/2]. To fix this, add pi to theta.
-0.05 + pi = color(blue)3.09
So, the trigonometric form is 1.24 cis 3.09, or 1.24 (cos 3.09 + i sin 3.09).
