Question

Если sin2a=0.5,то найдите sin^4+cos^4

Answer 1

${sin}^{4} \alpha + {cos}^{4} \alpha = {sin}^{4} \alpha + {cos}^{4} \alpha + 2 {sin}^{2} \alpha {cos}^{2} \alpha - 2 {sin}^{2} \alpha {cos}^{2} \alpha = {( {sin}^{2} \alpha + {cos}^{2} \alpha )}^{2} - 2 {sin}^{2} \alpha {cos}^{2} \alpha = 1 - 2 {sin}^{2} \alpha {cos}^{2} \alpha \\ \\ sin2 \alpha = 0.5 \\ 2sin \alpha cos \alpha = 0.5 \\ sin \alpha cos \alpha = \frac{1}{4} \\ {sin}^{2} \alpha {cos}^{2} \alpha = \frac{1}{16} \\ \\ 1 - 2 \times \frac{1}{16} = 1 - \frac{1}{8} = \frac{7}{8}$
ответ: 7/8.

Answer 2

$\sin^4\alpha +\cos^4\alpha =(\sin^2\alpha)^2 +(\cos^2\alpha )^2=(\sin^2\alpha)^2+2\sin^2\alpha\cos^2\alpha+\\\\+(\cos^2\alpha )^2-2\sin^2\alpha\cos^2\alpha=(\sin^2\alpha+\cos^2\alpha)^2-\frac{1}{2} \cdot4\sin^2\alpha\cos^2\alpha=\\ \\ =1^2-\frac{1}{2} \cdot2^2\sin^2\alpha\cos^2\alpha=1-\frac{1}{2} \cdot(2\sin\alpha\cos\alpha )^2=1-\frac{1}{2} \cdot(\sin 2\alpha )^2=\\\\=1-\frac{1}{2} \cdot(0,5)^2=1-0,5\cdot0,25=1-0,125=0,875$

ответ: $0,875$