I/D #3: Unit Q: Concept 1: Pythagorean Identities

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There are the reciprocal identities that I will constantly be referring to.

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These are the ratio identities that I will constantly be referring to.


1. Identities are proven facts or formulas that are always true. The Pythagorean Identity sin²x+cos²x=1 comes from using the Pythagorean theorem.  The Pythagorean theorem, instead of using a²+b²=c², we will replace the letters with x²+y²=r², which is the same exact thing when placed on a coordinate plane. Then we want the right side to equal 1 so we divide both sides by r² and we get(x/r)²+(y/r)²=1. When looking at what we now have, we can make to changes which we learned previously. First, we can change (x/r)² to cos² (cosine) because the equation for cosine on the unit circle is x/r. Also, we can change (y/r)² to sin² (sine) since that is the equation for sine on the unit circle. Now, to make sure that this is really identity, we must apply the unit circle to this to get the Let's plug in an angle of 45* to this identity. First, we find cos of 45 and sin of 45 which turn out to be √2/2. So we plug it into the Pythagorean Identity that we found. So it is going to be (√2/2)²+(√2/2)²=1. When we distribute the power and add both of them together, our answer becomes 4/4 which equal 1. Remember to put x or theta because that means that it can be any angle. Now we have one of our Pythagorean Identity which is cos²x+sin²x=q.

©Kelsea

©Kelsea

2. There are two remaining Pythagorean Identities, 1+tan²x=sec²x and 1+cot²x=csc²x. To obtain the first equation I mentioned, first divide both sides by cos²x. By doing so we get cos²x/cos²x+sin²x/cos²x=1/cos²x. Now when we look at our ratio identities and our reciprocal identities, we can see that some of the fractions we have created are equal to some sort of single identity: sin²x/cos²x=tan²x and 1/cos²x=sec²x. So we can replace the fractions with these single identities to get the equation 1=tan²x=sec²x.  We now have one last Pythagorean Identity to find. To do so, we divide by sin²x. Now our identity will be cos²x/sin²x+sin²x/sin²x=1/sin²x. Again, the fractions correspond with some of the ratio and reciprocal identities. cos²x/sin²x=cot²x and 1/sin²x=csc²x. Now replacing the fraction identity with a single identity, we get the last Pythagorean Identity which is cot²x+1=csc²x.

©Kelsea

References:

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