In the previous lessons, we manage to make very good progress with the development of complex numbers, though with the help of a great mathematician name Euler and his formula
However, inquisitive people as we all are, let us suppose for a minute that we did not have access to Euler's Formula at that time. Now this would be a problem because we needed to find another way to prove the result of the multiplication of two complex numbers.
NOTE: Most of the video is on proofs using trigonometry identities. If you are not interested in proofs, you perhaps want to move to the next section.
PROOF USING TRIGONOMETRIC IDENTITIES.1
May want to check out:
PROOF USING TRIGONOMETRIC IDENTITIES.2
FURTHER COMPLEX NUMBERS RESULTS
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VIDEO LECTURE #1
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LESSON
At this point, we have a good number of results, some of which was attained with the help of Euler’s Formula
. By using this formula, we can multiple complex numbers and in turn find out their magnitude and argument using laws of indices such as
which says that the product of However, let’s just say that for the moment, we need to prove the above results without Euler’s Formula. Honestly, I do not know what result came first but having another way to show the product We specify two complex numbers
and
then
This gives us the polar form of
In addition, by looking at the angle inside the cosine and sine functions, we can see that, A similar method can be used to calculate the magnitude and argument of the complex number I would like to conclude with proper notation to write the results.
It shouldn’t take the reader long to draw parallels with these results and the logarithm function. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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will give a resulting complex number which has a magnitude equal to the product of each number’s magnitude and an argument which is the sum of each number’s argument.
will undoubtedly be useful. And that is achieve using trigonometry identities. 
which we can immediately conclude that 
, the sum of the argument of 
and the argument of 
, is the argument of 
.
, which I will leave to the reader. You simply multiply the trigonometry identities as follows but use another addition formula to achieve the desired result.
