Pythagorean Identity of TrigonometryDate: 02/16/98 at 20:12:18 From: Jon Swaine Subject: sin^2 + cos^2 = 1 proof I noticed that most trig proofs I'm doing right now involve the identity sin^2 + cos^2 = 1. I was just wondering if anyone there could show me a proof of this useful identity. Thanks very much. Jon Swaine Kingston, Ontario, Canada Date: 02/16/98 at 22:56:55 From: Doctor Jaffee Subject: Re: sin^2 + cos^2 = 1 proof Hi Jon, I'd be happy to work through the proof with you. Get out a sheet of paper and draw a right triangle and label the two legs a and b. Label the hypotenuse c. Next, label the acute angle opposite side a with an A. Now, I assume you are familiar with the Pythagorean Theorem, which we can apply to your drawing and get a^2 + b^2 = c^2. Furthermore, by definition, sin A = opposite/hypotenuse = a/c and cos A = adjacent/hypotenuse = b/c. We are now ready for the proof. Since a^2 + b^2 = c^2, we can divide both sides of the equation by c^2. Our result is a^2/c^2 + b^2/c^2 = c^2/c^2, which is equivalent to (a/c)^2 + (b/c)^2 = 1. But that is just sin^2 A + cos^2 A = 1, and our proof is complete. You can see why some mathematicians refer to this identity as the Pythagorean Identity of Trigonometry. I hope this proof was clear, and good luck with your studies. -Doctor Jaffee, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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