Wednesday, March 19, 2014

I/D #3: Unit Q: Pythagorean Identities

Inquiry Activity Summary

1. Where does sin ^2x + cos^2x = 1 come from?

Let's go back to oh like maybe three weeks?? (I'm awful with time) where we first learned about the Unit Circle. In the Unit Circle we learned that the ratio for cosine is (x/r) and the ratio of sine is (y/r). Now let's reflect back on Pythagorean Theorem which is a^2 + b^2 = c^2,  but in context of the Unit Circle it would be x^2 + y^2 = r^2, but in the Unit Circle r always equaled 1 and in order to make this true and correct, we would divide by r^2 on both sides, leaving us with (x/r)^2 + (y/r)^2 = 1. But wait...(x/r) and (y/r) look similar...wait....isn't that cosine and sine? Why yes, yes it is. But they're squared. Because they are squared, we have resulted towards a Pythagorean Identity. Which cannot be "powered up"or "powered down", because it is a Pythagorean Identity it must always be squared and no power greater and no power less. Also because the Pythagorean Theorem is a proven fact and formula that is always true, it is called an identity. We can prove this by demonstrating one of the "Magic 3" ordered pair from the Unit Circle (30*, 45*, 60*). We'll use 60* now theta of 60* is (1/2, radical 3/2) with 1/2 being x and radical 3/2 being y. Since we want to prove our derivation of x^2 + y^2 = r^2 we're going to use cos^2 +sin^2 = 1, so (1/2)^2 + (radical 3/2) ^2 =1.

2.
In order to derive the remaining two Pythogorean Identities we must divide the whole thing by cos^2x to find the tangent derivation which will lead us to tan^2x + 1 = sec^2x

And then we divide the original by sin^2x to find the cotangent derivation leading us to 1 + cot^2x = csc^2x. 

Now you might be wondering how did we know some of this information? Well this information was found from our Unit Q SSS packet, information we should have, say it with me, MEMORIZED!!!






Inquiry Activity Reflection

1. The connections that I see between Units N, O, P, and Q so far are...

  1. The most obvious one would be the role of sine and cosine in this Unit Q from the Unit Circle and its properties which can be seen in the reciprocal identities which are similar to the inverse of these trig. functions from Unit O and P. 
  2. In Unit P Concept 3 when using the distance formula for Law of Cosines with SSS or SAS, one of the formulas is a^2 = c^2(sin^2A + cos^2A) - 2bcCosA + b^2 and we know that cos^2A + sin^2A = 1. 

2. If I had to describe trignometry in THREE words, they would be... 

mentally-draining (this counts as one word), stressful, and insightful. 

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