Inquiry Activity Summary:
1. The Pythagorean Theorem is an identity that can always be proven true no matter how much it is manipulated. It is evident in the Unit Circle when we take the sine and cosine of the 45º, for instance. The ordered pair would be (√2/2, √2/2) and the sin45 is √2/2 and the cos45 is √2/2. If we plug it into the Pythagorean theorem, it would be one equals one, which is true. Sin2x+cos2x=1 comes from the pythagorean theorem, which is x2+y2=r2. Next, you have to manipulate the pythagorean theorem so that there is only one "r". This is because in the unit circle, r is equal to one. To do this, you divide everything by r2. Once you have done this, you will get (x2/r2)+(y2/r2)=r. However, you can arrange that to make it look simpler: (x/r)2+(y/r)2=1. Now, you know that x/r is equal to cosine and y/r is equal to sine. As a result, you just substitute or replace it with cos and sin. Then, it would look like cos2x+sin2x=1.
2. The second pythagorean identity is 1+tan2x=sec2x. To get this, you divide the first identity by cos2x so that it will cancel and is equal to one. It will look like this: 1+(sinx/cosx)2=(1/cosx)2. We know that sine is equal to y/r and cosine is equal to x/r, so you can cancel out the r's and end up with y/x. This would mean that it's tangent, so you substitute it in. Next, 1/cosx is secant because cosine is x/r but it's the denominator, so you take the reciprocal of it. The third pythagorean identity: 1+cot2x=csc2x. To get this, you divide the first identity by sin2x so that the sin2x can cancel and equal one. So far it will look like this: 1+(cosx/sinx)2=(1/sinx)2. We know that x/r is cosine and y/r is sine, so we can cancel out the r's and get x/y, which is cotangent. 1/sinx is (1/ y/r), to get rid of the fraction you do the reciprocal which is r/y, and that would be cosecant.
Inquiry Activity Reflection:
1. "The connections that I see between Units N, O, P, and Q so far are..." they all relate to the Unit Circle and they incorporate the Pythagorean Theorem one way or another
2. "If I had to describe trigonometry in three words, they would be..." Ratios, "Unit Circle", FUN.
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