Polar Form of Imaginary Number w=8i - Undefined Tan(theta)

In summary, when converting a complex number to polar form, if the real part is 0, the angle can be immediately determined by plotting the point on the argand diagram. In this case, the angle is pi/2.
  • #1
trajan22
134
1
w=8i
I need to put this in polar form but how can i do this since this would be
w=8(cos(theta)+isin(theta))
I can't find the angles because tan(theta)=8/0
which of course is undefined. Is there something that I am doing wrong?
 
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  • #2
Nope. You haven't done anything wrong. tan(theta) simply isn't defined for certain angles. Which ones? The good news is that you can draw a picture and see immediately what the angle is. Hint, a "vector" to this point in the complex plane would lie completely along the imaginary axis. So what would theta be?
 
  • #3
So it could be either pi/2, or 2pi/3 but considering that 8 is a positive number then the angle must be pi/2 right?
 
  • #4
Why don't you just substitute pi/2 in w=8[cos(theta)+isin(theta)] and see which one gives you the correct ans?
 
  • #5
trajan22 said:
So it could be either pi/2, or 2pi/3 but considering that 8 is a positive number then the angle must be pi/2 right?

I think you mean it can be pi/2 or 3pi/2. You are correct that it must be pi/2. One way to see this is to plot the point on the argand diagram. 8i lies on the positive imaginary axis, and so the principal argument is the angle between the positive real axis and the positive imaginary axis, measured anticlockwise; this is equal to pi/2.
 

1. What is the polar form of the imaginary number w=8i?

The polar form of the imaginary number w=8i is 8i(cos(π/2) + i*sin(π/2)). This can also be written as 8i * e^(iπ/2), where e is the base of the natural logarithm.

2. How do you convert a complex number from rectangular form to polar form?

To convert a complex number from rectangular form (a+bi) to polar form (r(cosθ+isinθ)), you can use the following formula: r = √(a^2 + b^2) and θ = arctan(b/a). Then, plug in the values for r and θ into the polar form equation.

3. What does the term "undefined" mean in the expression "Undefined Tan(theta)"?

The term "undefined" in the expression "Undefined Tan(theta)" means that the tangent function is undefined for the given value of theta. In this case, it means that the value of tan(theta) cannot be determined because the angle theta is undefined or does not exist.

4. How is the polar form of an imaginary number related to its rectangular form?

The polar form of an imaginary number is related to its rectangular form through the use of the Pythagorean theorem and trigonometric functions. The magnitude of the imaginary number, or the value of r in the polar form, is equal to the square root of the sum of the squares of the real and imaginary components in the rectangular form. The angle, or value of θ in the polar form, is determined using trigonometric functions based on the real and imaginary components in the rectangular form.

5. Can the polar form of an imaginary number be used to calculate its powers?

Yes, the polar form of an imaginary number can be used to calculate its powers. To calculate the power of an imaginary number in polar form, you can use the following formula: (r(cosθ+isinθ))^n = r^n(cos(nθ)+isin(nθ)), where r is the magnitude and θ is the angle in the polar form. This can be simplified further using De Moivre's theorem: (r(cosθ+isinθ))^n = r^n(cos(nθ)+isin(nθ)) = r^n(cos(nθ)+isin(nθ)) = r^n(cos(nθ)+isin(nθ)).

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