1. Okay so we know that we are dealing with identities because the title is a given, yet what are identities? Well, an identity is basically any proven fact or formula of which brings the Pythagorean Theorem to mind considering that it is a formula that works and is always correct when put to use. Okay so let's start off with the Pythagorean Theorem. If we were given the terms x, y, and r then we would write out the formula like this: x^2+y^2=r^2, but we would tweak it a bit in order to have it equal to 1. We would divide by r^2, do we agree? Yes! Because that is pretty much the only way to equal an equation to 1, by dividing by whatever number the equation is equaled to. Then we would have (x/r)^2+(y/r)^2=1.
Now, if we go back to the unit circle, oh yeah I forgot to mention that the unit circle does come back to haunt haha, we know that cosine is x/r or in other words, adjacent over hypotenuse and sine is y/r or opposite over hypotenuse, so ..do we see anything? Hmm.. Yea! It matches to much of the Pythagorean Theorem! We would just have to square the two variables on the left, x and y, and then equal them to 1. In the end we could have cos^2theta + sin^2theta= 1. Remember that cosine is x and sin is y so in reality we just do some switching around and have fun with it and therefore sin^2x+cos^2x=1 is referred to as a Pythagorean Identity. To prove the theory with some examples and so you can see it visually look at the images below if let's say we did the magic 45 degree.
2. Now, to derive the identity with Secant and Tangent we would start off with the original sin^2x+cos^2x=1. The first thing we would do is divide everything by cos^2x and we would get sin^2x/cos^2x + cos^2x/cos^2x= 1/cps^2x and we can still simplify this further! We know that sin^2x/cos^2x is equal to tan^2x because tangent is opposite over adjacent (y/x). For the easy one, we know that anything over itself is 1 so cos^2x/cos^2x is 1 and then lastly we know that 1/cos^2x is the same thing as saying sec^2x. We simply are plugging in until it simplifies nicely. To conclude we have tan^2x+1=sec^2x. Yay!
To derive the identity with Co-secant and Cotangent we would start with the same original one, but divide by sin^2x. We know that after doing so the sines will equal 1 and cos^2x over sin^2x is the same thing as saying cot^2x so we would have 1+cot^2x= ...csc^2x since 1/sin^2x is csc^2x.
In order to see what is going on the picture below will hopefully help.
Inquiry Activity Reflection:
1. "The connection that I see between units N, O, P, and Q so far is...that everything in some sort of way comes together to conclude and involve the unit circle and triangles are involved as well.
2. "If I had to describe trigonometry in 3 words, they would be...ratios, substitutions, and strangely comprehensible.
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