I don't understand how to convert -3-4i to polar coordinates. I need to know how so I am ready for my trig final next year. Can someone please show me step by step how to solve this? Thanks

I'm sorry, I mean could you convert w= 1+sqrt3i to polar coordinates. I already did the other. My bad.

r^2 = 1^2+3 = 4

r=2

tanθ = y/x = √3
x and y are positive, so we're in QI
θ = π/3

1+√3 i = (2,π/3)

Sure! I can help you understand how to convert -3-4i to polar coordinates.

To convert a complex number to polar coordinates, you need to find its magnitude (also known as the modulus or absolute value) and its argument (also known as the angle).

Step 1: Find the magnitude (modulus):
The magnitude of a complex number is given by the formula:

|z| = sqrt(Re(z)^2 + Im(z)^2)

In this case, Re(z) is the real part of the complex number (-3) and Im(z) is the imaginary part of the complex number (-4i). Substituting these values into the formula, we get:

|z| = sqrt((-3)^2 + (-4)^2)
= sqrt(9 + 16)
= sqrt(25)
= 5

So, the magnitude (modulus) of -3-4i is 5.

Step 2: Find the argument (angle):
To find the argument (angle), you need to use inverse trigonometric functions to find the angle between the positive real axis and the line connecting the origin and the complex number.

The formula to find the argument (angle) is:

arg(z) = atan(Im(z) / Re(z))

In this case, Re(z) is -3 and Im(z) is -4. Substituting these values into the formula, we get:

arg(z) = atan((-4) / (-3))
= atan(4/3)

To find the angle in degrees, you can use a calculator or convert the result to degrees. So, the argument (angle) of -3-4i is atan(4/3) (in radians).

Step 3: Express the result in polar form:
To express the complex number in polar form, we write it as:

z = |z|(cos(arg(z)) + i sin(arg(z)))

In this case, |z| is 5 (the magnitude) and arg(z) is atan(4/3) (the argument). Substituting these values into the formula, we get:

z = 5(cos(atan(4/3)) + i sin(atan(4/3)))

This is the polar form of -3-4i.

Therefore, the polar coordinates of -3-4i are (5, atan(4/3)).

I hope this explanation helps you understand how to convert a complex number to polar coordinates. If you have any more questions, feel free to ask!