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CTK Exchange
jdawson
Member since Apr-2-11
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Apr-02-11, 10:05 PM (EST) |
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"Trigonometric proof of the Pythagorean Theorem"
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While I find Jason Zimba's proof interesting, I am puzzled why it'should have taken so long for someone to cast a proof of the Pythagorean Theorem in trigonometric form. To see that the geometric definitions of the trigonometric functions are well-defined requires similarity theory. But granting that, it is easy to modify Euclid's similarity proof (VI,31) without resort to the sum or difference identities. For let c be the hypotenuse of right triangle ABC, and draw the altitude from the opposite angle, C. Then
c = a cos B + b cos A, and similar triangles yield that
a/(a cos B) = c/a and b/(b cos A) = c/b. Replacing c in the last two equations by the right member of the first equation and then cross-multiplying givesa^2 = a^2 (cos B)^2 + ab cos A cos B and
b^2 = ab cos A cos B + b^2 (cos A)^2 . Hence a^2 + b^2 = a^2 (cos B)^2 +2ab cos A cos B + b^2 (cos A)^2
= (a cos B + b cos A)^2 = c^2 . This would seem to me to be a "purely trigonometric proof". Am I missing something?
JWD |
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