SOLUTION: Write the complex number in trigonometric form r(cos θ + i sin θ), with θ in the interval [0°, 360°). -12 - 16i

Plot the point (-12,-16).
Draw a line from it to the origin.
Draw another line from it perpendicular to the x-axis.
That makes a right triangle with horizontal side the 
x-coordinate of (-12,-16) which is x=-12, and vertical
side the y-coordinate of (-12,-16), which is y=-16.
We label the length of the the hypotenuse as "r".  
Draw an arc to indicate the rotation of angle θ starting 
at the right side of the x-axis and swinging around
counter-clockwise to the slanted line.  

 

We calculate the value of r by the Pythagorean theorem:








We calculate the angle θ by first calculating its reference
angle, which is the angle inside the triangle indicated by
the small green arc.  We use only the absolute values (positive
values) of the sides.

We can take our pick of which trig ratios to use,

sine of the green reference angle = opposite/hypotenuse = y/r = 16/20
cosine of the green reference angle = adjacent/hypotenuse = x/r = 12/20
tangent of the green reference angle = opposite/adjacent = y/x = 16/12

Use the inverse trig functions on your calculator with of any of the 
positive ratios and get the green reference angle as 53.13010235°.

So the red angle θ is that value + 180° or 233.13010235°.

So we substitute r=20 and θ=233.13010235° in



and get



Edwin