1. Where does sin^2x + cos^2x=1 come from to begin with?
To start off, an "identity" is proven facts and formulas that are ALWAYS true. The Pythagorean Theorem is an identity because it has been proven that in a triangle, for instance, a^2 + b^2= c^2. For this unit, the Pythagorean Theorem can be consist of x,y, and r similar to that of a, b, and c according to the Unit Circle in quadrant I.
Because a^2 + b^2=1, we divide r^2 on both sides. We can use our knowledge from the Unit Circle to determine what is (x/r)^2 and (y/r)^2. From there, we have found our first Pythagorean Identities.
Cos^2x + sin^2x=1 is referred to as a Pythagorean Identity because we used the Pythagorean Theorem to find it. Basically, we substituted variables and connect that to what we have learned.
To be sure if this Pythagorean Identity is true, we can use the angles (30, 45, or 60 degrees) from the Unit Circle and substitute those values. This picture shows an example of using the 60 degrees.
2. How to derive the two remaining Pythagorean Identities from sin^2x + cos^2x=1
The picture above demonstrates the various steps of how to get tan^2x + 1 = sec^2x. First, you will have to divide both sides by cos^2x. Then, with your memory of the ratio identities, reciprocal identities or Pythagorean Identies, you would substitute it into the equation. The identity with Secant and Tagent is tan^2x + 1 = sec^2x.
The picture above shows the various steps of how to get 1 + cot^2x = csc^2x. You divide sin^2x on both sides. Then, using your memory, plug in what you know. At last, you should be able to find the identity for Cosecant and Cotangent.
INQUIRY ACTIVITY REFLECTION
1.THE CONNECTIONS I SEE WITH UNIT N, O, P AND Q SO FAR ARE ratios that we use again from the Unit Circle and how we use quadrant I to know what variables to plug into the equation that we are trying to find.
2. IF I HAD TO DESCRIBE TRIGONOMETRY IN THREE WORDS, THEY WOULD BE difficult but fun.
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