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

1. Where does sin^2 x+cos^2 x=1 come from to begin with?

To begin with. you should know that an "identity" is "a proven fact that is always true". Therefore, we assume that a Pythagorean Identity, which is the equation seen above, is called such because that equation can definitely be proven as a fact that is always true. Remember that the Pythagorean Theorem using variables x, y, and r is x^2+y^2=r^2. Now doesn't this look familiar? Yes, we did get it from the a^2+b^2=c^2 equation we've learned before, but notice that just by changing the variables to x, y, and r gives us the exact equation we learned that is of a unit circle.

Now let us play with this equation a little. Who knows what amazing discovery we will come across. If we wanted to set the Pythagorean Theorem (which we just found out is similar to the unit circle equation) equal to 1 as shown in the Pythagorean Identity above, what would we do to it? Oh, of course! Divide everything by r^2 because r^2/r^2 is equal to 1. We are left with x^2/r^2+y^2/r^2=1. You may think, what kind of mess is this? Well, do not fear, Math For Cheese Buckets, like you and me, is here to make it clear! First of all, you should know that the equation can be re-written as (x/r)^2+(y/r)^2=1. 

Moreover, the ratio of cosine is (x/r) and the ration of sine is (y/r). Wait, we've seen these before. Yes, a ton of times in the last 2 units, but wait a sec, we just saw them right now, in the re-re-written equation of the Pythagorean Theorem. We can look at the equation in the previous paragraph and switch the (y/r)^2 with sine^2 x and the (x/r)^2 with cosine^2 x. Guess that we did come across a discovery. We just derived the Pythagorean Identity from the Pythagorean Theorem. 

To show that this identity is true, we will choose one of the "Magic 3" ordered pairs from the unit circle.

• 30 degrees= (radical 3 over 2, 1/2)
• 45 degrees= (radical 2 over 2, radical 2 over 2)
• 60 degrees= (1/2, radical 3 over 2

I will show 30 degrees and 45 degrees (know that 30 degrees is similar to 60 degrees, just switched).

• 30 degrees: (radical 3 over 2)^2 + (1/2)^2 = 1. You cancel out the radical with the squared and end with just the 3 on the top of the first fraction. You square the 2 and end up with a 4 at the bottom of the first fraction, so you have 3/4. For the second fraction you square the 1 and still end with a 1 in the top of that second fraction, and you square the 2 to end up with a 4, so you have 1/4. (3/4)+(1/4)=1.
• 45 degrees: (radical 2 over 2)^2 + (radical 2 over 2)^2 = 1. You can cancel out the radical with the squared for the top of both fractions to end up with a 2 on top of both fractions. Then, square the 2 at the bottom of both fractions and end up with a 4 at the bottom of both fractions: (2/4)+(2/4)=1.

2. Show and explain how to derive the two remaining Pythagorean Identities.

sine^2 x + cosine^2 x = 1. You divide the equation by cosine^2 x first. We know that sine x divided by cosine x equals tangent x by the Ratio Identities. So, we can "power up" through our understanding that it will be the same thing if we squared the sine and cosine we can square the tangent. The first part of the equation becomes tangent^2 x. Obviously, cosine^2 divided by itself is equal to 1. The second part of the equation is 1. On the right side of the equation 1 divided by cosine^2 x will become secant^2 x, because we know by the Ratio Identities that 1 divided by cosine x is secant x, therefore we can again "power up" to show that secant x will become secant^2 x. Our final equation: Tangent^2 x + 1 = secant^2 x.

The picture below can give you a visual. Please ignore the bottom part. Just pay attention up until the equation that is squared.





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Now we divide the equation,  the one at the beginning of the top paragraph, by sine^2 x. Obviously, sine^2 x divided by itself is 1. Through the Ratio Identities we know that cosine x divided by sine x equals cotangent x. Again, we can "power up" to show that cosine^2 x divided by sine^2 x equals cotangent^2 x. On the right side of the equation we show that 1 divided by sine x equals co-secant x through the Ratio Identities, so once again we "power up" to know that 1 divided by sine^2 x will now equal co-secant^2. Our final equation: 1 + Tangent^2 x = Co-secant^2 x.

The picture below will give you the equations I am deriving the above work from.







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