I/D #3: Unit Q - Pythagorean Identities

INQUIRE ACTIVITY SUMMARY

1.) Where does sin^2x+cos^2x=1 come from to begin with? You should be referring to the Unit Circle ratios and the Pythagorean Theorem in your explanations. 

The Pythagorean Theorem is considered an "identity" (def: a proven fact or formula that is always true). The Theorem a^2+b^2=c^2 can also be written with different variable like x,y, and r to get: x^2+y^2=r^2. In order to get the equation equal 1, r^2 would need to be divided on both sides to get: (x/r)^2+(y/r)^2=1. From this new equation we can get information using our knowledge from the Unit Circle. If we remember, x/r is the trig ratio for cosines while y/r is the ratio for sines. The equation can be written as: cos^2(x)+sin^2(x)=1. This is referred to as a Pythagorean Identity since it was first derived from the Pythagorean Theorem, which is an identity that was simply rewritten but never lost its value. To prove that this equation is an identity and always true, we can take one of the "Magic 3" ordered pairs from the Unit Circle. For example, the ordered pair for a 45 degree angle is (radical2/2, radical2/2). When the ordered pair is plugged into the equation: (radical2)/2)^2 + (radical2)/2)^2, the answer WILL result to get 1. 

2.) Show and explain how to derive the two remaining Pythagorean Identities from cos^2(x)+sin^2(x)=1. Be sure to show step by step. 

     a) If the equation were to be divided by cos^2, the equation would look like:  cos^2(x)/cos^2(x)+sin^2(x)/cos^2(x)=1/cos^2(x). The first part (highlighted in blue) will cancel out to equal 1 since both numerator and denominator are the same. The second part (highlighted in green) can be substituted by a ratio identity of tan^2(x). The third part (highlighted in light blue) can be substituted by a reciprocal identity of sec^2(x). In the end, the derivation will result as 1+tan^2(x)=sec^2(x).

     b) If the equation were to be divided by sin^2, the equation would look like:  cos^2(x)/sin^2(x)+sin^2(x)/sin^2(x)=1/sin^2(x). The first part (highlighted in purple) can be substituted by a ratio identity of cot^2(x). The second part (highlighted in orange) cancels out to equal 1 since both numerator and denominator are the same. The third part (highlighted in red) can be substituted by a reciprocal identity of csc^2(x). In the end, the derivation will result as cot^2(x)+1=csc^2(x).

INQUIRE ACTIVITY REFLECTION

"The connections I see so far in Unit N, O, P, and Q so far are..." the repeating use of the trig functions of sine, cosine, tangent, cosine, cosecant, and cotangent from the Unit Circle AND the use of triangles.

"If I had to describe trigonometry in THREE words, they would be..." triangles, triangles, and triangles!...Just kidding; it would be complex, overwhelming, and triangles.