I/D #3: Unit Q: Pythagorean Identities

1. Where does sin²x + cos²x  = 1 come from?

http://i1.ytimg.com/vi/o-fAx_96lgw/maxresdefault.jpg

Well, let's start by defining this as the main Pythagorean Identity. So what is an identity? They are proven facts and formulas that are always true, so we can go ahead and manipulate the expressions and equations. The equation that we can manipulate (not redefine) is the Pythagorean Theorem because it  is also an identity that will help us help us derive sin^2x+cos^2x=1.
So let's back up a little. The Pythagorean Theorem is a^2+b^2=c^2. However, when referring to a unit circle, the Pythagorean Theorem is written as x^2+y^2= r^2.

http://www.contracosta.edu/legacycontent/math/Pythagoras.htm

This is how the Pythagorean Theorem is drawn on the unit circle, with r=1. So how do we get r^2 to equal 1? We simply do divide it all by r^2 (both sides)

x^2 + y^2 =r^2
                               >>>  (x/r)^2 + (y/r)^2 = 1    We cancelled off r^2
r^2     r^2   r^2

Now, take a closer look at the fraction and what do you see? Anything familiar, perhaps the ratio for cosine and sine on the unit circle?? Yes? Yes!
Based on one of previous units, we found out that the ratio for cosine is x/r and the ratio for sine is y/r. Notice how they are written the same way, except that they are both squared. So we can just replace the ratios by the trig function name. Therefore, we get: sin^2x+cos^2x=1
This helps prove the reason why the Pythagorean Theorem is an identity since it can be manipulates, but the facts and formula is always the same even though a little rearranged.
Let's actually prove that this works though. Since we are talking about the unit circle, we can use one of the "Magic 3" pairs from the Unit  Circle. Let's use 45 because it has different x and y values.
The ordered pair is :(√2/2)>>>> Substitute them in the equation:  (√2/2)² + (√2/2)² = 1. (√2/2)²
When we simplify it, we get (1/2) + (1/2)=1, so this helps us prove that then identity is true!

Now we can proceed to find the next two Pythagorean Identities:
We can first find the tangent derivation by dividing the main identity by cos^2/x.
sin^2x   +   cos^2x    =    1
                                                                >>  tan²x + 1 = sec²x.                         The cos^2x cancel to equal 1
cos^2/x       cos^2/x     cos^2/x                                Then we find the rest by using the ratio/reciprocal identities


It is very important to remember the ratio identities in order to make your life easier and know where we get our substitutions!

http://www.sosmath.com/CBB/viewtopic.php?t=41908

Now we can continue to find cotangent derivation by dividing everything by sin^2x
sin^2x   +   cos^2x    =    1
                                                                >>  cot ²x + 1 = csc²x                The sin^2x will cancel to equal 1
sin^2x         sin^2x       sin^2x                                 Then we find the rest by using the ratio/reciprocal identities


Inquiry Activity Reflection:
1. The connections that I see between Units N, O, P, and Q so far are that the unit circle and the trig functions will continue to reappear and help define most of the trig functions and anything that relates to it even though equations are always being rearranged. Also, I learn the significance of sin and cos because all trig functions eventually relate back the them. It is almost like the universal trig function. 
2. If I had to describe trigonometry in THREE words, they would be... interconnected, confusing,and mind-blowing.