-3 + 4i in polar form in pi

the magnitude is 5 from pyth theorm. Now the angle. Using i the positive real as the reference axis, the angle from it is PI/2 + arctan4/3

To express -3 + 4i in polar form in terms of π, we need to find its magnitude and angle.

1. Magnitude (r):
The magnitude (or modulus) of a complex number is the distance from the origin to the point representing the complex number in the complex plane. We can use the Pythagorean theorem to find the magnitude:
|r| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5

2. Angle (θ):
To find the angle, we need to determine the direction (or angle) of the vector from the positive real axis (x-axis) to the complex number -3 + 4i.

We can use the arctan function (or inverse tangent) to find the angle. The arctan function (tan^(-1)) gives the angle corresponding to a given ratio of the sides of a right triangle. In this case, we can take the ratio of the imaginary part to the real part of the complex number (-3 + 4i).

θ = tan^(-1)(4/(-3))

Using a calculator, we find the value of θ to be approximately -0.93 radians (-53.13 degrees). However, since we want the angle in terms of π, we convert it to the equivalent form:

θ = -0.93 radians = -0.93 * π/π = -0.93π

Therefore, the polar form of -3 + 4i in terms of π is 5 cis(-0.93π) or 5e^(-0.93πi).