z = 5 + 12i
5 is the real term which forms the horizontal leg of the right triangle formed.
12i is the imaginary term where 12 forms the vertical leg of the right triangle formed.
the hypotenuse is called r.
r = sqrt(5^2 + 12^2) = sqrt(169) = 13
theta is the reference angle formed whose vertex is at the origin.
tan(theta) = 12/5
theta = arctan(12/5) = 67.38013505.
that's the reference angle which is the equivalent angle in the first quadrant.
the real part is on the x-axis.
the imaginary part is parallel to the y-axis.
because the real part is negative and the imaginary part is positive, the angle is in the second quadrant.
the equivalent angle in the second quadrant is 180 minus 67.38013505 = 112.6198649 degrees.
the equation, in polar form is:
z = r * (cos(theta) + i * sin(theta))
that becomes:
z = 13 * (cos(112.6198649) + i * sin(112.6198649)
that should be your solution.
you can convert it back to rectangular form by simplifying the equation to get:
z = 13 * (cos(112.6198649) + i * sin(112.6198649) is equal to:
z = 13 * (-.3846153846 + i * .9230769231) whish is equal to:
z = -5 + i * 12) which is the same as:
z = -5 + 12i
an online polar coordinate calculator shows the following:
this calculator can be found at https://www.intmath.com/complex-numbers/convert-polar-rectangular-interactive.php