convert the numbers to polar form and then divide.leave the answer in polar form. 10-10i/3root3+3i
To divide complex numbers and leave the answer in polar form, you can follow these steps:
Step 1: Convert the complex numbers to polar form.
Let's convert the first complex number, 10-10i, to polar form:
To find the magnitude (r) of 10-10i, we can use the Pythagorean theorem:
r = √(10^2 + (-10)^2) = √(100 + 100) = √200 = 10√2
To find the argument (θ), we can use the inverse tangent function:
θ = arctan(-10/10) = arctan(-1) = -π/4 or -45 degrees (quadrant IV)
Therefore, 10-10i in polar form is: 10√2 ∠ -π/4 or 10√2 ∠ -45 degrees.
Now, let's convert the second complex number, 3√3 + 3i, to polar form:
To find the magnitude (r) of 3√3 + 3i, we can use the Pythagorean theorem:
r = √((3√3)^2 + 3^2) = √(27 + 9) = √36 = 6
To find the argument (θ), we can use the inverse tangent function:
θ = arctan(3/3√3) = arctan(1/√3) = π/6 or 30 degrees (quadrant I)
Therefore, 3√3 + 3i in polar form is: 6 ∠ π/6 or 6 ∠ 30 degrees.
Step 2: Divide the complex numbers in polar form.
To divide two complex numbers in polar form, divide their magnitudes and subtract their arguments:
(10√2 ∠ -π/4) / (6 ∠ π/6)
Dividing the magnitudes: (10√2) / 6 = (5√2) / 3
Subtracting the arguments: (-π/4) - (π/6) = -3π/12 - 2π/12 = -5π/12
Therefore, the result of the division in polar form is: (5√2) / 3 ∠ -5π/12 or (5√2) / 3 ∠ -75 degrees.