Question

Чему равно $sin^4(a)+cos^4(a)$ , если $sin(a)+cos(a) = \frac{\sqrt{2}}{2}$

Answer 1

$\sin a+\cos a=\sqrt{2}/2\Rightarrow (\sin a+\cos a)^2=1/2;\ \sin^2 a+\cos^2 a+2\sin a\cos a=1/2;$

$1+2\sin a\cos a=1/2;\ 2\sin a\cos a=-1/2;\ \sin a\cos a=-1/4.$

$\sin^4 a+\cos^4 a=((\sin^2 a)^2+2\sin^2 a\cos^2 a+(\cos^2 a)^2)-2\sin^2 a\cos^2 a=$

$(\sin^2 a+\cos^2 a)^2-2(\sin a\cos a)^2=1^2-2(-1/4)^2=1-1/8=7/8.$

Ответ: $\frac{7}{8}$