Re: Sum of squares

On 19-05-2007 0:29, chrizm7@xxxxxxxxx wrote:

Hmm okay, I see the counter example. But my book has the following:

a^2 + b^2 = 1

It then uses cos^2(t) + sin^2(t) = 1

and says there is some angle t such that

a^2 = cos^2(t)

b^2 = sin^2(t)

How is this justified?

Since a^2 + b^2 = 1, a^2 <= 1 and so -1 <= a <= 1. On the other hand,
cosine is continuous, cos(0) = 1, and cos(pi) = -1, and therefore there
is some _t_ in [0,pi] such that cos(t) = a. Then

sin^2(t) = 1 - cos^2(t) = 1 - a^2 = b^2.

Best regards,

Jose Carlos Santos
.