Formulas you need to know:
- z=x+yi
- z=rcos(theta)+rsin(theta)i
- z=rcis(theta)
- z=square root of x^2+y^2
When multiplying complex numbers:
- If it is a rectangular problem, you always use foil.
- If it is a polar problem, you always multiply r and add theta.
Example 1: 5+2i
- First, you have know if it is in rectangular form or polar. It is in rectangular form.
- Now, since you know it is in rectangular form, you know you will have to find r and theta.
- Use the formulas from above ^^^^^. To find r, you have to use the r=square root of x^2+y^2 formula.
- Put the numbers in the formula and solve: R=square root of 5^2+2^2=square root of 25+4=square root of 29. (i always equals 1)
- Now, you have to find theta by using the formula Theta=tan^-1(y/x).
- Put the numbers in the formula and solve: Theta=tan^-1(2/5)=21.801 and then you have to put it in quadrant one and then find the other answer which is 201.801.
- You can now put your answers in their final form: square root of 29cis21.801 and -square root of 29cis201.801.
Example 2: 8cis240 degrees
- Like the problem above, you have to know which form this equation is in. It is in polar form.
- Since it is in polar form, you now know that you have to find x and y.
- You have to use the formulas x=rcos(theta) and y=rsin(theta).
- Since the degrees in the original problem is not between 0-90, we have to find a reference angle. 240-180=60.
- Now you can place the numbers in the formulas.
- X=8cos60=8(1/2)=4
- Y=8sin60=8(square root of 3/2)=6.928
- Your final answer is z=4+6.928
-Amber :)
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