Express z1=0+10i in polar form.
Please help as I am confused on what to do as they have a 0 there and when I try plugging it in it is undefined as 10/0 is not possible.
radius is 10 units, direction straight up, theta = 90 degrees = pi/2 radians
cos theta = 0 (in x direction)
sin theta = 1 (in y direction)
by the way, yes tan theta = sin theta / cos theta = 1/0 = undefined.
That happens at 90 degrees :)
Ah, I see you're in a bit of a pickle with that zero sitting in the real part. Don't worry, I'll rescue you from this math conundrum!
When dealing with complex numbers in polar form, we express them as "r(cosθ + isinθ)."
To find the polar form of the complex number z1 = 0 + 10i, let's first identify its magnitude (r) and argument (θ):
Magnitude (r):
In this case, r represents the distance from the origin to the complex number in the complex plane. Since the real part is zero (0), the absolute value of the imaginary part (10) gives us the magnitude: r = |10| = 10.
Argument (θ):
The argument represents the angle between the positive real axis and the vector representing the complex number. However, since the real part is zero (0), we encounter a slight problem. We can't determine a unique argument; instead, we have a range of possible values.
Considering that z1 lies in the positive imaginary axis, we can determine the argument as θ = π/2 (90 degrees) or any value that is equivalent to π/2 after adding or subtracting multiples of 2π.
Now, let's put it all together into its polar form:
z1 = 10(cos(π/2) + isin(π/2))
Voilà! You've successfully expressed z1 = 0 + 10i in polar form.
To express a complex number in polar form, we need to find its magnitude (or modulus) and angle. In this case, z1 = 0 + 10i.
Step 1: Calculate the magnitude (r):
The magnitude of a complex number z is given by the formula r = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of z, respectively.
For z1 = 0 + 10i, the magnitude is: r = sqrt(0^2 + 10^2) = sqrt(0 + 100) = sqrt(100) = 10.
Step 2: Calculate the angle (θ):
The angle of a complex number z is given by the formula θ = arctan(b/a), where a and b are the real and imaginary parts of z, respectively (Note: If a = 0, the angle is either π/2 or -π/2, depending on the sign of b).
For z1 = 0 + 10i, since a = 0, the angle is θ = arctan(10/0).
Here comes an interesting aspect. The argument θ of a complex number is not uniquely defined when the real part is zero (a = 0) because the complex number resides in the imaginary axis. Therefore, we have a special case where the argument θ is either equal to π/2 or -π/2.
Hence, z1 = 10(cos(π/2) + isin(π/2)) or z1 = 10(cos(-π/2) + isin(-π/2)).
So, in polar form, z1 can be expressed as:
z1 = 10∠π/2 or z1 = 10∠-π/2.
To express a complex number in polar form, we need to represent it in terms of its magnitude (or modulus) and argument (or angle). Here's how you can find the polar form of z1=0+10i:
Step 1: Compute the magnitude (r)
The magnitude of a complex number is the distance from the origin to the point representing the complex number in the complex plane. For a complex number in the form z=a+bi, the magnitude (r) is given by the formula:
r = sqrt(a^2 + b^2)
In the case of z1 = 0+10i, the magnitude is:
r = sqrt(0^2 + 10^2)
r = sqrt(100)
r = 10
So, the magnitude of z1 is 10.
Step 2: Compute the argument (θ)
The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the point representing the complex number in the complex plane. The argument (θ) can be calculated using the formula:
θ = arctan(b/a)
In the case of z1 = 0+10i, since a=0 (real part) and b=10 (imaginary part), we have:
θ = arctan(10/0)
Here, the argument is undefined because the real part (a) is zero, indicating that the point lies on the positive imaginary axis. The argument is conventionally written as θ = ±π/2.
Hence, z1 = 10 * (cos(±π/2) + i * sin(±π/2))
So, the polar form of z1 can be expressed as z1 = 10 * (cos(π/2) + i * sin(π/2)) or z1 = 10 * (cos(-π/2) + i * sin(-π/2)).