вправа 15.27 гдз 10 клас математика Істер 2018

Умова:

Відповідь - ГДЗ:

1) sinα + cosα = √2 cos(α - π/4)
sinα + cosα = sinα + sin(π/2 - α) =
= 2sin π/2 cos (α - π/2 + α)/α =
= 2sin π/4 cos(α - π/4) =
= 2 • √2/2 cos(α - π/4) = √2 cos(α - π/4);
2) (cosβ+cos5β) / (sin5β-sinβ) = ctg2β
(cosβ+cos5β) / (sin5β-sinβ) =
= (2cos3βcos2β) / (2sin2βcos3β) = cos2β / sin2β = ctg2β;
3) (sin15α-sinα+sin7α) / (cos15α+cosα+cos7α) = tg7α
(sin15α-sinα+sin7α) / (cos15α+cosα+cos7α) =
= (2sin7αcos8α+sin7α) / (2cos8αcos7α+cos7α) =
= (sin7α(2cos8α+1)) / (cos7α(2cos8α+1)) = tg7α;
4) (sinх+sin3х+sin5х+sin7х) / (cosх+cos3х+cos5х+cos7х) = tg4х
(sinх+sin3х+sin5х+sin7х) / (cosх+cos3х+cos5х+cos7х) =
= (2sin2хcosх+2sin6хcosх) / (2cosхcos2х+2cos6хcosх) =
= (2cosх(sin2х+sin6х)) / (2cosх(cos2х+cos6х)) =
= (2sin4хcos2х) / (2cos4хcos2х) = sin4х/cos4х = tg4х.