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Pythagorean Identity of Trigonometry

Date: 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 

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
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Associated Topics:
High School Trigonometry

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