I/D#3: Unit Q- Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:

• First of all, we must make the connection that the Pythagorean Theorem is an identity. An identity is "a proven fact or formula that is always true". So in the PY, a^2+b^2=c^2, it's a proven statement. On a graph, instead of using "a,b,c" we use "x,y,z" which we can refer back when we were graphing special right triangles within the Unit Circle. So how do we get from x^2+y^2=r^2 to the pythagorean identity of sin2x+cos2x=1? We simply just divide r^2 on both sides to the x^2 and y^2. So now we have (x/r)^2+(y/r)^2=1. This is the moment where things just connect together and we can all turn on our lightbulbs! We see the ratio x/r and y/r and we know the trig functions that go with it. So when we plug that in we get sine from (y/r) and cosine from (x/r)! Since it's squared, we get the PI of sin2x+cos2x=1.
  
• How can we be sure that it actually works? We can use one of the Magic 3 pairs from the Unit Circle! Below is an example using 60 degrees:
  

• Here are the remaining Pythagorean Identities are derived from sin2x+cos2x=1
  
  

• ID w/ secant and tangent
  When we derive secant and tangent from our base Pythagorean Idenity, sin2x+cos2x=1, there are two things we know from memorizing the chart! We know that to get tangent, we would need sin2x/cos2x. Also, to get secant, we know the recipricol identity will be 1/cos2x. So how do we get this? Simply divide cos2x from everything and we are left with our Pythagorean ID of tan2x+1=sec2x!
  
  
  
• ID w/ cosecant and cotangent
  Similar to the previous problem, there are something we already know from the chart! To get cosecant and cotangent, we must divide sin2x from everything. From here, we know that cos2x/sin2x is the ratio identity for tan2x. As for 1/sin2x, we know that is the recipricol identity for csc2x. After we plug in the identities to the corresponding one, we get our Pythagorean ID of 1+cot2x=csc2x.

INQUIRY ACTIVITY REFLECTION:

• The connections that I see between Units N, O, P, and Q so far are that the Unit Circle from Unit O will always be our base/foundation. From it, we can use trig functions that we found from the special right triangles in the circle. The SRT also used the Pythagorean Theorem! So from Unit Q back to N, it all connects in the way that the identities are made up from ratio, recipricol, and pythagorean.
• If I had to describe trigonometry in THREE words, they would be tricky, complicated, "oOoOoOoOh" (when I understand the concept)!