I/D3: Unit Q - Pythagorean Identities

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 a^2+b^2=c^2. Since a, b and c are variables, we can replace them with any variable of our preference. But, according to the ratios, we use x, y and r, manipulating the theorem into x^2+y^2=r^2. We then have to divide the equation by r^2 to make it equal 1 and we would end off with (x/r)^2+(y/r)^2=1. Using our knowledge from unit O, we remember the trig ratio for cosines is x/r and y/r is the trig ratio for sines. If we simply substitute those variables for cosine and sine, the equation would be cos^2(x)+sin^2(x)=1. Proving this theorem to be an identity, we use one of the "Magic 3" ordered pairs from the Unit Circle and plug it in to our new equation. In example, the ordered pair for a 45 degree angle according to the unit circle is (radical2/2, radical2/2). Plugging it into the equation, we get (radical2)/2)^2 + (radical2)/2)^2, which ends up equaling 1, like every angle on the unit circle should.

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. 

To show the left over 2 Pythagorean identities, we just have to manipulate the formula.

If we divide the equation by sin^2, the equation would be: cos^2(x)/sin^2(x)+sin^2(x)/sin(x)=1/sin^2(x). cos^2(x)/sin^2(x) would be substituted by cot^2(x). sin^2(x)/sin(x) would equal 1 since both top and bottom are the same. 1/sin^2(x) would be substituted by csc^2(x). In the end, the equation would end up being cot^2(x)+1=csc^2(x).

Now that we did sine, we do cosine. If we divide the equation by cos^2, the equation would be cos^2(x)/cos^2(x)+sin^2(x)/cos^2(x)=1/cos^2(x). cos^2(x)/cos^2(x) would equal 1 since top and bottom are the same. sin^2(x)/cos^2(x) would be substituted by the ratio tan^2(x). 1/cos^2(x) would be substituted by sec^2(x). In the end, the equation would end up being 1+tan^2(x)=sec^2(x).

INQUIRE ACTIVITY REFLECTION:

"The connections I see so far in Unit N, O, P, and Q so far are..." the use of the unit circle and the trig functions along with the ordered pairs from the unit circle.

"If I had to describe trigonometry in THREE words, they would be..." triangles, ratios and the unit-circle.