Question

Determine a distância entre os pontos A (cos a, sen a) e B (sen a, cos a).

Answer 1

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Solução!

Formula da distância!

$d(A,B)= \sqrt{(xA-xB)^{2}+(yA-yB)^{2}}\\\\\\\\ d(A,B)= \sqrt{(cosa-sena)^{2}+(sena-cosa)^{2}}\\\\\\\\ d(A,B)=\\\\ \sqrt{(Cos^{2}b-2cosa.sena+sen^{2}a )+(sen^{2}a-2sena.cosa+cos^{2}a )}\\\\\\\\ d(A,B)=\\\\ \sqrt{(Cos^{2}b-2cosa.sena+sen^{2}a )+(sen^{2}a+2sena.cosa+cos^{2}a )}\\\\\\\\ d(A,B)= \sqrt{(Cos^{2}b+sen^{2}a )+(sen^{2}a+cos^{2}b )}\\\\\\\\ d(A,B)= \sqrt{(2sen^{2}a +2cos^{2}a )}$

$\boxed{Sen^{2}a+cos^{2}a=1 }\\\ ~~~~~~~~~~~~~~~~~~~~~~~~\Downarrow~\\\ d(A,B)= \sqrt{2(sen^{2}a +cos^{2}a)}\\\\\\\\ d(A,B)= \sqrt{(2(1)}\\\\\\\\ \boxed{d(A,B)= \sqrt{2}}$


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