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

$sin^4alpha +cos^4alpha =(sin^2alpha)^2 +(cos^2alpha )^2=(sin^2alpha)^2+2sin^2alphacos^2alpha+\\+(cos^2alpha )^2-2sin^2alphacos^2alpha=(sin^2alpha+cos^2alpha)^2-frac{1}{2} cdot4sin^2alphacos^2alpha=\ \ =1^2-frac{1}{2} cdot2^2sin^2alphacos^2alpha=1-frac{1}{2} cdot(2sinalphacosalpha )^2=1-frac{1}{2} cdot(sin 2alpha )^2=\\=1-frac{1}{2} cdot(0,5)^2=1-0,5cdot0,25=1-0,125=0,875$

Ответ: $0,875$