Words of Encouragment

 

 

The Work

My favorite literary character is Sherlock Holmes. I have read the complete works of this series by Sir Arthur Conan Doyle at least twice and made a point of visiting 221b Baker Street on a vacation to London last year. I have also enjoyed the movies with Robert Downey Jr. and Jude Law. However, I must make special emphasis on my love for BBC's Sherlock series, which transports Holmes and Watson to present day London, but I digress.

 

What is commonly remembered of Sherlock Holmes is his power of "deduction" (actually it's induction), which is his ability of grabbing the minute details of a subject and arriving at a conclusion. For example: Holmes would take a look at a person's clothes and tell their occupation, errands of the day, mode of transportation, degree of anxiety and town of residence. He would do so by noticing things like tips of mid in the shoes, the pattern their clothes are crumpled, a ticket or stub protruding from a pocket, a button in the shirt not completely fastened, the know of a tie...etc. front hese actions people say That Sherlock Holmes is a brilliant man.

 

What most people never realize is that Sherlock Holmes spent most of his time on study. He would famously analyze 140 different type of Tobacco ash. Therefore when he got on a scene and saw Tabacco Ash he could say what kind it was. In other words Sherlock Holmes was brilliant because he would do the work (and had an extraordinary brain that could make sense of all he learned and appy it as needed). Lacking the later I have taking to weeks to do the former with my trigonometry.

 

As promised, I took all this week to study up on trigonometric identities and the values of sin, cos and tan at various degrees. I had to do this because I felt I had reached a wall in my studies that was making it difficult for me to understand the newer concepts I was being given in Pre-calculus. This situation was creating a mental block that was giving me a chance to doubt my abilities to carry on my goal of learning calculus.

 

I am happy to say that, as I suspected and sincerely hoped, a serious attempt to memorize and analyze trigonometric properties and identities has helped me a lot.

 

I started with the basic stuff. It was taking me too long to grasp angles quoted in radians. Therefore, I spend a day with degree and radian conversions. I did not want to only memorize that π/6, π/4, π/3, π/2 and π were 30deg, 45deg, 60deg, 90deg and 180deg respectively. I wanted to understand it and see it. As it turns out, a quick drawing of an unit circle is very handy for this, as is knowing your multiples of 90, 180, 270 and 360.

 

Then I progressed with the value of trig functions at various degrees. The degrees I studied were of course the "nice degrees" as the people at mathmistakes.info call them. These are angles were sin, cos and tan (along with their reciprocal functions) have exact values that are easy to remember.

 

I cannot thank the people behind mathmistakes.info enough since they had in their site the very tool I needed to help me study. It was a chart with all the trig values in the easy angles for all quadrants. However, their chart has all the values hidden from view until you ask it to show them. It does not have a button for hide all or show all, so finding a pattern would be difficult, but it is a great way to test your knowledge. You can find the chart here.

 

Doing "THE WORK" helped me join together all the concepts I had studied so far in trigonometry. This might sound absurd since I have been doing trig quizzes for my pre-calc class the last 3 weeks, but it's true nevertheless.

 

All weeks prior, I've been relying on my memory and the examples the professors gave in class. It is very common that quiz questions can be answered using the exact process the Profesor used in class, and that was the case with my homework. Therefore, as long as I followed what the professor did step by step I would get a correct answer without any deep understanding of the process used. In othet words, I had gone through a lot of knowledge in pre-calculus class, but that knowledge didn't stay fresh in my mind for long. Case in point: reference angles.

 

A few weeks back I studied reference angles, which are acute positive angles that are coterminal to other angles and the x axis. This is a fancy way of saying that if you have an angle that is 150^o, it's reference angle is the smallest angle formed from the terminal side of that angle to the x axis. Therefore the reference angle of 150^o is 30^o. It is a lot easier to understand when you see it in the figure opposite.

The homework for that section was very simple and straight forward. I felt I was getting a free pass. If I had paid attention to the importance of reference angles in Trig values, it would have saved me a lot of headaches.

 

You see, It was not until I was studying the relationship between trig values and the unit circle that I finally understood the importance of reference angles. Once I had learned sin values from 0° to 90°, I went over to the other quadrant and thought to myself: Since trig functions are cyclical, sin of 120° must equal sin of 30° since 90°+30°=120°. When I went over to a reference chart to check this, I found out I was incorrect.

 

Sin of 30° was not equal to sin of 120°, sin of 30° is equal to sin of sin of 150°. That threw me for a loop until I put the values in an unit circle. Then it all made sense. Not only did I find a pattern for the values of sin, cos and tangent, but I also understood that the value of all of them is equal to the value of their reference angles before taking their sign value into consideration. Therefore since 30° is the reference angle of 150°, 210° and 330°, sin is 1/2 (or -1/2 depending of the quadrant) in all those angles.

 

It was an excellent week of study and discovery. Of course these discoveries are small compared to Sherlock Holme's deductions. If fact they are small in terms of what every high school student already knows of trigonometry. But for me that are huge and important because I found them on my own. Nothing can compare to the thrill of figuring things out for yourself.

 

Let me know how I'm doing or if I got something wrong.

 

The Wall

For a while, I have been able to see it from a distance. Every day I would check to see if I saw it moving closer, knowing that if it did, I would be in trouble. When I returned this week from vacation I was not surprised to see it right in front of me. I am talking about the wall. That seemingly insurmountable force that wants me to stop my quest to learn calculus by making it look daunting, foolish and, worst yet, insignificant.

 

I was expecting to have lingering doubts about how consequential my desire to learn calculus is. I knew a day would come when I would ask myself: "What am I doing? This is ridiculous. I am just going to waste my time with this." The posts on words of encouragement were a weekly reminder against this.

 

What I was not expecting was the attack on my intellectual ability to actually do this. When the wall appeared this week, it had an inscription that says: "You cannot understand any of this anymore. It will just get worse, I promise." It's hard to argue with the inscription when it rings so true. The last two weeks I have been struggling to truly grasp the concepts I have been reviewing. They are mainly pre-calculus properties and transformations. I have a hard time using radians and relating them to trig graphs and the unit circle.

 

Part of me knows that the reason i am struggling is that I have not done THE WORK. I have not consciously gone throughout the process of memorizing all the trigonometric properties and, more importantly, the values of sin, cos and tan for special angles in the unit circle. Still, a small but potent voice, tells me that I am just rationalizing the real problem: you are incapable of learning this. And for this week I have been to much of coward to find out which voice was right.

 

Until today.

 

Today I will begin THE WORK. I will set out to store in my long term memory all the trigonometric properties I can. I see cue cards and pop quizzes in my future. I am confident that the work I put in these days will be invaluable when I start with Calculus.

 

Wish me luck. I have a wall to turn down. Now, where is my sledgehammer?

 

Yes, today is π day 3.14

And to think I almost missed it when I have been battling with inverse trigonometric functions all week!

 

Check out this site for all things PI. http://www.piday.org/

I promise to post soon...vacations can put a hold on Math blogs!

 

On logarithms and funny notations

I remember having trouble with logarithms in high school because I had to do a transformation that hurt my head to understand. Before reviewing this section of pre-calculus, I consulted my memory bank's folio on logs. It read: "Logs will ask you to do stuff to convert them that will feel unnatural." I could not remember what it was exactly, but I could see the numbers bouncing around each other rearranging themselves. As I reviewed the chapter I remember what that is.

 

Logarithms are just the accepted notation to answer this question: to what power do I need to raise x number to give me answer y? In other words, I might want to know to what power should I raise 2 to get 64. In exponential form I would write 2^x=64.

 

However, in logarithmic form we would grab the x exponent throw it across the equal sign, then take the 2 shrink it, append it to the word "log" and finally take the 64 across the equal sign to join log₂. At the end we should have something like this: log₂64=x.

I hope you can see why it blew my mind as a teenager.

 

The Log transformation got me to think about math notation, or the way we represent mathematical concepts. I mean, if you think about it, someone at some point had to decide that the letter x would make a good place holder for a number we don't know; or that a cross would signify addition and a a horizontal line subtraction. I bet those first proponents had to explain what the symbols meant:

"Dear colleagues, whenever you see this notation of two parallel horizontal lines "=" it means that the terms on either side of "=" are equal"

I can almost hear the collective "Ahhh"s of fellow mathematicians reveling in not having to write the word "equal" anymore. I am of course being a little facetious. I am bet the introduction of the equal sign did not happen quite that way...but come on, I bet the person who came up with the radical sign (√) for square roots got a long of grief from the long division enthusiasts.

Getting back to logarithms, the logarithmic and exponential, as best as I can explain from what I understood, equations are crucial to describe rapid growth (or decline). Imagine you had a magic ball that would split into two magic balls every hour, then the new balls would also split every hour and so on. How many balls will you have in 24 hours? Pick a number in your mind and make it high. If you number is 16.7 million balls, you are right on the money.

The exponential or logarithmic graph has a segment rises or falls very quickly and another that gets progressively smaller but never touches the point it gets close two. That, if I remember correctly is an asymptote. A basic exponential graph has the x-axis as an asymptote, the logarithmic graph usually has the y-axis.

If you look at the graphs of log₂(x)and 2^x , the first is logarithmic and the second exponential, they will be mirror images of each other along the y=x line. This happens because in the logarithmic equation, the y values you get are the power you need to raise 2 in order to get the x values, while in the exponential one the y values you get are the result of raising 2 to the x power.

The cousera course videos keep telling me that exponential and logarithmic equations are important to calculus. For the time being, I guess I must take their word for it.

 

If anyone thinks I am missing something, let me know.

 

'Till next time.