Number Sense, Common Sense, and the Unit Circle

Posted by: | Posted on: January 22, 2012

Math may mean nothing to you, so I’ll give the main point first: the use of authority by parents that comes from a set of rules to be memorized creates a(n) often resentful, but sometimes placid servant-child.  Authority that comes from an understanding of what’s going on creates a future-leader.

Now onto the math.  Please keep reading for a little bit, all of you math-phobic men and mamas:

I hated the unit circle.  I knew how to fill in all those parts…but at the time, I didn’t know why it was important.  But then I discovered a very gifted young lady that had absolutely no sense of trigonometric number sense.  So, there was no way for them to determine that cos(200) was really close to -1.  Or in one particular case,  sin(276) is a lot closer to 0 than to -1, so if they may have their calculator set to radians instead of degrees.

This young lady had been taught the rules to memorize the relationships of a certain reference triangle, and to use it to solve problems.  But when we got to Calculus, and we had to come up with an idea based on the understanding of sine and cosine, she was lost.  She could answer a question, but she couldn’t understand the solution.  Therein lies the problem with “because I said so” – authoritarian parenting.  Imagine this applied to family life.  “Don’t let strangers in.  Either don’t answer the door, or tell them your mom is in the shower”.  This is perfect for a 4 year old.  However there may come a time when the stranger at the door is there to help him (a policeman or a fireman clearing in the case of an evacuation).  What a conflict for the poor young man.  Worse still are the moral implications.  “Please lie sometimes, but never lie to Mom…because….because…she’s bigger than you”.  Better to be told to identify the stranger, and what he/she needs before determining what to say….and if there is a loss for words, tell them that he/she can’t let them in until their mom’s at the door.   No lies, more understanding.

Mathphobics may stop reading now….back to the math:


The above image illustrates the idea of the unit circle.  This is a circle with radius 1.  When you drop a line to the x axis it creates a right triangle.  With this right triangle, there are an infinite number of possibilities for x and y, however the ones to remember are the ones that form special right triangles.  Special right triangles are the same ones used as reference triangles:



Please notice that the second triangle comes from drawing a diagonal in a unit square.  Anyway, apparently this young lady, and everyone else working from that book had been told to memorize that relationship, and then use SOHCAHTOA to determine the value.   This helped them get the ANSWER.  However, understanding the solution requires this understanding:


From which we can then see that cosine is the x value of the unit circle, and sine is the y value.  Furthermore, when they draw in the special angles of the “reference triangles” (30, 45, 60) they get this beauty:


From which you can see that cosine of 276 better be close to 0, or something is messed up about that calculator.  And its okay to follow a fireman out of building, because he may be a stranger that is saving your life. AND, mommy doesn’t lie, or tell me to lie, so I shouldn’t lie to her.


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