Multiple Methods

I am so sorry that this post has taken so long to come out. I had difficulty logging into my Squarespace account and then school really came in full force. Here is the body of my next post. Here it is though!!

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This past year in math, my understanding of the “correct“ way to solve a problem has drastically changed. When situation I thought were algebra dependent have different conditions, my faulty assumption was the the problem was entirely unsolvable. Recently, however, I learned that what can be solved with algebra can also be solved with geometry, vector computations, and even using the imaginary plane. I will give a few examples of how methods of finding solutions changed my understanding of problem solving as a whole.

This first comes with integrals. Last year, we memorized all of the integral “rules.“ What the integral of sin(x), a^x, 1/x, and the various arc identities. We had limits, however. For example, using simple manipulation techniques, it was impossible or extremely tedious to solve the integral of sqrt(a^2 +/- f(x)^2 ). Such integral would not appear on an assessment as we thought it could not be solved by hand. The first integral technique we learned, however, was the use trig substitution for such problems. When setting up the portion inside of the square root as the side lengths of a triangle, you could find a trig relationship for the problem. Then, using the Pythagorean theorem, the square root could be solved and the entire problem was not broken down into a way our memorized rules could apply. An example is shown below:

But, lets say that even with this method, you can’t find the integrals of the trig functions. Or, let’s say that this entire function is multiplied by e^x. Then what?

Well, you can of course do by parts 2-3 times only increasing the size of the integral. And while that sometimes might lead to the right answer, there are many places for error along the way and it may just loop back. This is when a new method of solving was introduced to us, something that rattled my understanding of trigonometry as a whole.

First, we redfined what complex numbers were as an introduction. A complet number written as “x + yi” could be sketch out as a triangle in the complex plane. Just like with how radians are definined, this triangle was part of a unit circle. This was represented by e^i+theta. See below for the visual relationship:

After defining a complex number as such, the complex number (represented by e to the i, theta) could be written in terms of sin and cos. The maginitude e^i theta is the same as the unit circle, 1. This is also the same as the sqrt(cos^2(x) + sin^2(x)). This means that the complex number can be written as cos(theta) + i sin(theta). All of these relationship are written below to clearly see how these new definitions are achieved.

After this simple redefinition of complex numbers, we changed out definition of what sin(x) and cos(x) were. We could revert back to terms of “e“ instead of using trigonometry. This means that sin and cos are oficially represented in the complex plane. See below for the two official definitions:

With this new definition of cos and sin, integrals that were once unsolvable in the non-complex plane can be solved. One such example when this is useful is in the template below:

Using the memorized identities, this might seem impossible to solve. Although there are ways to take the integral of powers, it would involve longer computational work to work out the products of the two powers of e without using the complex plane. Substituting the identies we found for e to a power along with cos and sin, we can get the formula below:

This was just the most recent example of what I though math was limited too or inefficient in solving could be simplified into an incredibly easy process as long as the correct approach is applied. What I think is possible in math is always changing, including when using the complex plane could be applicable. I hope that as I continue studying this every changing subject, my understanding will continue to be broken and rebuilt as I learn to tackle problems in different ways.