Thursday, March 20, 2014

I/D #3: Unit Q Concept 1: Pythagorean Identities

Inquiry Activity Summary:

 1). An identity is a proven fact or formula that is always true. Pythagorean Identities are proven to be true and are one type of identities that we are studying in this concept. They can be easily derived through our knowledge of the Pythagorean theorem and the Unit Circle. The following will show precisely how I have done just that.

  • So basically what I did was I wrote out the Pythagorean theorem and then made it equal to 1 by dividing everything evenly by r squared. I then used what I knew of the trig functions, specifically the ratios of sine and cosine to rewrite the equation. The end result was the first Pythagorean identity.
  • The next thing I did was I tested the truth of the equation. I used what I knew of the Unit Circle to get a point (in this case the point for 45 degrees) and plugged in the cosine and sine values of that point into the equation. The equation met expectations, meaning that the equation is indeed an identity.
2). The other two main identities of the Pythagorean Identities can be derived by

simply by dividing by either cosine squared or sine squared from the first identity.
  • After dividing by cosine, the cosines will cancel into a 1 and using the ratio and reciprocal identities we find that we are left with tangent and secant. The result is the second Pythagorean identity.
  • After dividing by sine we find that like the other one, something cancels into a one and the trig functions change (this time into cotangent and cosecant). The result this time is the third main Pythagorean identity.

Inquiry Activity Reflection:

1). The connections I see between Units N, O, P, and Q so far are that the identities are related to the Unit Circle (Unit N) in that we use an r value of 1 for the Pythagorean theorem and we use the sine and cosine of a point on the unit circle to test the equation and that we use the relationships of the various trig functions to do the derivations. We also use the Pythagorean in the other derivations we have done for the various other units.
2). If I had to describe trigonometry in three words they would be: integrated, complex, and understandable.

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