I/D#3- Unit Q- Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:
1) sin^2x+cos^2x=1 is a Pythagorean Identity which means that it can be a proven fact and formula that is always true. So to start deriving it we use the Pythagorean Theorem. Pythagorean Theorem is always a^2+b^2=c^2 but we have learned different when dealing with unit circle. We know that the leg are the x-axis and the one going up and down is the y-axis and we know that the hypotenuse is r when it comes to the unit circle. So know we just switch it really instead of having a^2+b^2=c^2 we can a new one that is x^2+y^2=r^2 and we want to get to sin^2x+cos^2x=1. To do that we divide everything by r^2 and we should get x^2/r^2+y^2/r^2=1 it then turns to (x/r)^2+(y/r)^2=1. Right here we should notice something about the variables and the ratios. We know that cosine is x/r and we know that sin is y/r so, then we plug it into our formula and (y/r)^2 is sin^2x and (x/r)^2 is cos^2x so know we put it together and we should get that identity which should be sin^2x+cos^2x=1

I choose the 60º from the "Magic 3" in the Unit Circle to show that this identity is true. The picture is below to see that it can be proven.

2) OTHER PYTHAGOREAN IDENTITIES

    The first pythagorean identity contains Secant and Tangent. The only way we could get that is by dividing by either sine or cosine. In this case we divided by cosine so that we can get what we want which is tangent and secant. So we divide by cos^2x and for sin^2x/cos^2x you should know well you should memorize that it is always going to be tan^2x. for the second part IT SHOULD NOT BE ZERO IT SHOULD BE ONE.The third part we use the reciprocal identity and it should be sec^2x. So when you put it all together you get tan^2x+1=sec^2x

The other pythagorean identity contains Cosecant and Cotangent. Since we divided the last one by cosine lets divide this one by sine now. we notice for part 1 that IT IS NOT GOING TO BE ZERO IT IS GOING TO BE ONE DONT FORGET. The second part you should know by memory that it should be cot^2x. The last part is the reciprocal so the reciprocal of sin is csc so it is csc^2x so your final answer should be 1+cot^2x=csc^2x

INQUIRY ACTIVITY REFLECTION:

“The connections that I see between Units N, O, P, and Q so far are…” The connections are that we are still using the trig functions cosine sine tangent .etc. and the references of the unit circles.

“If I had to describe trigonometry in THREE words, they would be…” Stress, overwhelming RATIOS