Answer:
Step-by-step explanation:
Recall that
<u>Part A:</u>
We are just squaring a binomial, so the FOIL method works great. Also, recall that .
<u>Part B:</u>
The magnitude, or modulus, of some complex number is given by .
In , assign values:
<u>Part C:</u>
In Part A, notice that when we square a complex number in the form , our answer is still a complex number in the form
We have:
Expanding, we get:
This is still in the exact same form as where:
- corresponds with
- corresponds with
Thus, we have the following system of equations:
Divide the second equation by to isolate :
Substitute this into the first equation:
This is a quadratic disguise, let and solve like a normal quadratic.
Solving yields:
We stipulate and therefore is extraneous.
Thus, we have the following cases:
Notice that . However, since , two solutions will be extraneous and we will have only two roots.
Solving, we have:
Given the conditions , the solutions to this system of equations are:
Therefore, the square roots of are:
<u>Part D:</u>
The polar form of some complex number is given by , where is the modulus of the complex number (as we found in Part B), and (derive from right triangle in a complex plane).
We already found the value of the modulus/magnitude in Part B to be .
The angular polar coordinate is given by and thus is:
Therefore, the polar form of is: