Monday, January 28, 2013

All Students Take Calculus in SohCahToa

Mnemonics saved my skin more than a few times in school, be it Please Excuse My Dear Aunt Sally which is used to remember the order to solve operations (parenthesis, exponent, multiplication, division, addition, subtraction), or My Very Energetic Mother Just Served Us Nine Pizzas to remember the order of the planets from the sun. The last one worked very well until Pluto was demoted...now Mom Just Serves Us Nachos. Talk about a downgrade.
In life after school, Mnemonics are now often used to create secure passwords. I read about two ways to create stronger passwords with this technique: one is before the fact and the other after the fact. Before the fact means that you take a memorable phrase (either universally memorable or better yet personally memorable) and create a password using the first letter of each word in the phrase. For example: Hickory dickory dock,the mouse ran up the clock,the clock struck one,the mouse ran down,hickory dickory dock becomes hddtmrutctcsotmrdhdd. You could even substitute "one" for 1.
To create a mnemonic device after the fact means that once you get a secure password, you then create a phrase or technique to remember it. Here I used the Norton Secure Password Generator to create this one: NadEbr2v. My mnemonic for it would be Nadia and dear Ernest bought roses too violet.
In trigonometry I ran across two mnemonic devices, one is SohCahToa, which helps you remember the angles Sine, Cosine and Tangent relate to. Sine is Opposite over Hypotenuse; Cosine is Adjacent over Hypotenuse; Tangent is Opposite over Adjacent.
 
The other one is All Students Take Calculus, and it is used to remember which trig formulas have positive signs in the unitary circle, the concept of which I'll try to summarize out of my review.
 
As I understand it an unitary circle is one that has a radius of 1 and its center is in the (0,0) coordinate of a Cartesian plane. In trigonometry, this unitary circle is used to study the functions of sine, cosine, tangent and their counterparts. Since the unitary circles has a radius of 1, if we use that radius to create a right triange with the x axis, that trianangle and any other right triangle we create with a point on the unitary circle circumnsference, will have a hypotenuse of 1.
Therefore, as per the Pythagoras theorem, a^2 + b^2 would equal 1. Using SohCahToa we know that sine and cosine are the lenght of the opposite side and adjacent side of a triangle respectively, divided by the lenght of the hypotenuse. Since the hypotenuse is always 1 in the unitary circle the sine and cosine of any angle made in the unitary circle equal the lenght of the "legs" of that triangle. I think I did a better job explaining it in the picture here.
 
The Unitary Circle can also be used, I learned, to get the important points of a sine, cosine and tangent graph. What I mean by important points, are the points that distinguish those functions and that help people analyze them.
 
For example, imagine an unitary circle as a clock and substitute 12 o'clock for π/2, 9 o'clock for π, 6 ο'clock for 3/2 π and 3 0'clock is 0. Here we will make a mental image of a right triangle in that circle. If that triange has no height, does it have and opposite side? No, but there is point in the unitary circle there that is (1,0). Therefore at 0, sine is 0 but cosine is 1, which means the sine graph will start at 0 and the cosine graph at 1. If we take the point π/2 (12 o'clock), we can say that triangle has no base but there is a point (0,1) there. At that point the cosine graph is 0, but sine is 1. Following the same rules sine is 0 at π and cosine is -1 just as sine would be -1 and cosine would be 0 at 3/2 π. The full circle is completed at 2π where sine is again 1 and cosine is 0. When you see it graphically it looks like this:
Check out this video here keeping in mind that π is approximately 3.14 and 2π is around 6.28.
 
I found my review of the unitary circle really fun. Let me know if I have made any mistakes here by leaving a comment.
 

4 comments:

  1. Not necessarily a Mnemonic but a very useful visual cue for memorizing the values of Sin, Cos, and Tan at 0,30,45,60 and 90. If you can memorize one column then Cos is Sin backwards.
    Sin in degrees at {0,30,45,60, 90) is {0,1/2, sqrt(2)/2, sqrt(3)/2, 1} cos is the values backwards {1, sqrt(3)/2, sqrt(2)/2, 1/2, 0} and then Tan is just sin/cos. You substitute degrees for their equivalent in rads

    ReplyDelete
    Replies
    1. Thanks for the tip. I'll be sure to use it in my final coming up in 3 weeks!

      Delete
  2. Another visual and memorization cue for memorizing Sin Cos and Tan in degrees at 0,45, and 90. On the Unit Circle, Sin is the Y axis and Cos is the X axis, at 0 degrees you are on the X axis so the answer is 1 and Sin is at its furthest away from the Y axis so the answer is 0. at 90 degrees you are on the Y axis so Sin is 1 and cos is at it's furthest away from the x axis to the answer is 0.

    No helpful tricks for memorizing the value at 45 degrees but Cos and Sin have the same Value at 45 degrees which is sqrt(2)/2

    Tan is again just Sin over cosine so we have at 0 degrees 0/1 = 0, at 45 degrees, a value over itself which is 1, and at 90 degrees 1/0 which is undefined.

    ReplyDelete