Pythagorean Identites Essay

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Pythagorean Identities Definition of the Pythagorean Identities There are three Pythagorean Identities. These three identities are derived from dividing the Pythagorean Theorem—a2+b2=c2 by a2, b2, and c2. In doing this, the trigonometric functions can be used. Using the triangle diagram below, here are the definitions of each identity: sin2 + cos2 = 12 + 2= 1Sin is equal to opposite (a) over hypotenuse (c) and cos is equal to adjacent (b) over hypotenuse (c).|1 + tan2 = sec21 + 2 = 2Tan is equal to opposite (a) over adjacent (b) and sec is equal to hypotenuse (c) over adjacent (b).|1 + cot2 = csc21+ 2 = 2Cot is equal to adjacent (b) over opposite (a) and csc is equal to hypotenuse (c) over opposite (a).| Proof for ( - = 0) Steps |Justification| - = 0|This is the original equation| - = 0|First, a common denominator must be found, in order to be able to subtract the problem. This is achieved by taking the denominator of each side and multiplying it on the opposite side of the subtraction sign, in both numerator and denominator. | - = 0|Next, the problem is simplified. The problem is multiplied in the appropriate places. | - = 0|The problem is further simplified in the highlighted portion.| - = 0|In this step, becomes because sin2 + cos2 = 1. This is so because in order to isolate sin2 cos2Is subtracted from both sides. This is equivalent to . (This step is a Pythagorean identity, as discussed above.)| = 0|Now subtraction can take place. There is a common denominator and everything is simplified in both fractions. When is performed, the difference is zero. | 0=0|This fraction becomes zero. When a fraction has a 0 in the numerator, the fraction is equal to zero. This is so because there are no parts in the fraction.

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