Question

если
$\cos( \alpha ) - \sin( \alpha ) = 0.2$
то найдите
cos³A－sin³A=

Answer 1

$\cos\alpha -\sin\alpha=0,2\\\\ (\cos\alpha -\sin\alpha)^2=0,2^2\\\\ \cos^2\alpha-2\cos\alpha\sin\alpha+\sin^2\alpha=0,04\\\\ (\cos^2\alpha+\sin^2\alpha)-2\cos\alpha\sin\alpha=0,04\\\\ 1 -2\cos\alpha\sin\alpha =0,04\\ \\ -2\cos\alpha \sin\alpha =0,04-1\\ \\ -2\cos\alpha\sin\alpha =-0,96 \\ \\ \cos\alpha \sin\alpha =0,48$

$\cos^3\alpha -\sin^3\alpha =(\cos\alpha -\sin\alpha )(\cos^2\alpha +\cos\alpha \sin\alpha +\sin^2\alpha )=\\ \\ =(\cos\alpha -\sin\alpha )(\cos^2\alpha -2\cos\alpha \sin\alpha +\sin^2\alpha+3\cos\alpha \sin\alpha )=\\ \\ =(\cos\alpha -\sin\alpha )((\cos\alpha -\sin\alpha )^2+3\cos\alpha \sin\alpha) =\\ \\ =0,2\cdot(0,2^2+3\cdot0,48)=0,2\cdot(0,04+1,44)=0,2\cdot1,48=0,296$