Write z=-6+6i in polar form
To convert from rectangular form to polar form, we use the following formulas:
magnitude (r) = sqrt(x^2 + y^2)
angle (theta) = tan^-1(y/x)
In this case, we have:
x = -6
y = 6
So:
r = sqrt((-6)^2 + 6^2) = 6*sqrt(2)
theta = tan^-1(6/-6) = -45 degrees or -pi/4 radians (since the point is in the third quadrant)
Therefore, z = 6*sqrt(2) * e^(-i*pi/4) in polar form.
write x=9-3*sqrt3 i in polar form
To convert from rectangular form to polar form, we use the following formulas:
magnitude (r) = sqrt(x^2 + y^2)
angle (theta) = tan^-1(y/x)
In this case, we have:
x = 9
y = -3sqrt(3)
So:
r = sqrt(9^2 + (-3sqrt(3))^2) = sqrt(108) = 6sqrt(3)
theta = tan^-1((-3sqrt(3))/9) = -30 degrees or -pi/6 radians (since the point is in the fourth quadrant)
Therefore, x = 6sqrt(3) * e^(-i*pi/6) in polar form.
To write a complex number in polar form, we need to express it in terms of its magnitude or modulus (r) and angle or argument (θ).
Given z = -6 + 6i, we can find its polar form as follows:
Step 1: Find the magnitude (r):
The magnitude of a complex number z = a + bi is given by:
|r| = √(a^2 + b^2)
In our case, a = -6 and b = 6. Plugging these values into the formula:
|r| = √((-6)^2 + 6^2)
|r| = √(36 + 36)
|r| = √72
|r| = 6√2
Step 2: Find the argument (θ):
The argument of a complex number z = a + bi is given by:
θ = arctan(b/a)
In our case, a = -6 and b = 6. Plugging these values into the formula:
θ = arctan(6/(-6))
θ = arctan(-1)
θ ≈ -0.7854 radians (in the fourth quadrant)
Now we have both the magnitude (r) and argument (θ) of the complex number.
The polar form of z = -6 + 6i is:
z = 6√2 * cis(-0.7854)