I/D #3:Unit Q Concept 1: Using Fundamental Identities to Simplify or Verify Expressions

INQUIRY SUMMARY ACTIVITY

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

J

An identity is a proven fact and/or formula that is ALWAYS true. The Pythagorean Theorem being itself an identity, as it is a proven statement, that is portrayed in a right triangle within the first quadrant of a unit circle. With using the Pythagorean Theorem, in this case, we use the variables x, y, and r not a^2+b^2= c^2. Instead of it being a, b, and c they would be replaced by x,y,r. 

As worked out in the image above all we had to do to get the Pythagorean Theorem equal to 1 is divide by r^2. By dividing by r^2, we got a ratio used/seen in the unit circle of x/r. Which turns out to be the ratio for cosine.  Another important thing seen is that for the term being added we got y/r, which represents sine when speaking in terms of the unit circle. If we were to substitute those for cos and sin we get: cos^2x + sin^2x = 1. With the substitution we have achieved our Pythagorean identity, through the use of Pythagorean theorem.

I chose one of the "Magic 3" ordered pairs from the unit circle to show that this statement is true. The ordered pair used was the 60* angle, which has coordinates of (1/2, rad3/2).

2. How to derive the two remaining Pythagorean Identities from sin^2x + cos^2x=1

The image above shows that how to derive the identity involving Secant and Tangent. First we have to divide everything by cos^2ø, which would cancel out the cos^2ø making it equal 1. Then we would substitute this ---------> sin^2ø/cos^2ø by tan^2ø. To finish it off we substitute this ---->  1/cos^2ø by this sec^2ø. The answer ending up looking like this: tan^2ø + 1 = sec^2ø. 

In this image the work is shown of how to derive the identity involving Cosecant and Cotangent. The first step taken is dividing everything by sin^2ø. Next we would cancel out sin^2ø and it would be left equal to one. Afterwards we are left with cos^2ø/sin^2ø on one side, what we do with that is substitute it with cot^2ø. Next we would substitute ---> 1/sin^2ø by csc^2ø. Leaving us with our final answer of: 1 + cot^2ø = csc^2ø. 

So now we have been able to tackle the task of deriving each of our three Pythagorean identities. ( tan^2ø + 1 = sec^2ø, 1 + cot^2ø = csc^2ø, and cos^2x + sin^2x = 1)