1. cos2x+sin2x=1 is derived from the unit circle. The pythagorean theorem is x^2+y^2=r^2. In the picture below I demonstrate how the pythagorean theorem relates to cos2x+sin2x=1, since the ratio for sine is y/r and for cosine its x/r we can also substitute that into the Pythagorean theorem and then divide everything by r2 to make it equal to one. This is another method for figuring out how cosine and sine are derived from the Pythagorean theorem.
2. We can also derive other forms of the equation sin2x+cos2x=1:
What I did in the first example is divided everything on both sides by sin2x. We then discovery that the division of sin2x leads to some ratio and reciprocal identities that change the equation to cotx and cscx. For step 3 I just substituted the identities into the equation.
For the second example I started off by dividing by cos2x. This also resulted in a ratio and reciprocal identities. Last step I substituted the new identities into the equation.
INQUIRY ACTIVITY REFLECTION:
1.The connections I see between Unit N, O, P, and Q so far are that many different equations can be derived from the unit circle and they can be right triangles or non right triangles.
2.If I had to describe trigonometry in three words, they would be Unit Circle, Triangles, and SOHCAHTOA.
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