How do you convert 1+(3i) to polar form?

1 Answer
Jan 14, 2017

(r,θ)=(2,120)

Explanation:

For a complex number in the form
XXXa+bi
the radius is given by the Pythagorean Theorem as
XXXr=a2+b2
In this case
XXXr=(1)2_(3)2=1+3=4=2

The angle can be calculated using an inverse trig function and adjusting to place the point in the proper quadrant.

In this case 1+3i is in Quadrant 2 of the complex plane.

Using the arctan function gives an angle in (90,+90)
so it will be necessary to add 180 to shift the angle into Q 2.
XXXθ=arctan(31)+180=120

(In this particular case, we have one of the standard triangles and the angle could easily be inferred with a quick sketch. )