вправа 22.60 гдз 11 клас алгебра Істер 2019
Вправа 22.60
Умова:
Розв'яжіть рівняння:
1) cos3х + cos5х = cosх;
2) cos2х = 1 + cos4х.
Розв'яжіть рівняння:
1) cos3х + cos5х = cosх;
2) cos2х = 1 + cos4х.
Відповідь ГДЗ:
✔ 1) cos3х + cos5х = cosх
cos3х + cos5х - cosх = 0 \begin{equation} 2cos\frac{5x+3x}{2}\cdot cos\frac{5x-3x}{2}- \end{equation} \begin{equation} -cosx=0 \end{equation} 2cos4х • cosх - cosх = 0
cosх • (2cos4х - 1) = 0
cosх = 0 2cos4х - 1 = 0
х = π/2 + kπ 2cos4х = 1
k ∈ Z cos4х = 1/2
4х = ±π/3 + 2kπ
х = ±-π/12 + πk/2
k ∈ Z;
✔ 2) cos2х = 1 + cos4х
cos2х - cos4х - 1 = 0
cos2х - (cos22х - sin22х) - (sin22х + cos22х) = 0
cos2х - cos22х + sin22х - sin22х - cos22х = 0
cos2х - 2cos22х = 0
cos2х(1 - 2cos2х) = 0
cos2х = 0 1 - 2cos2х = 0
2х = π/2 + kπ 2cos2х = 1
х = π/4 + kπ/2 cos2х = 1/2
k ∈ Z 2х = ±π/3 + 2kπ
х = ±π/6 + kπ, k ∈ Z.
✔ 1) cos3х + cos5х = cosх
cos3х + cos5х - cosх = 0 \begin{equation} 2cos\frac{5x+3x}{2}\cdot cos\frac{5x-3x}{2}- \end{equation} \begin{equation} -cosx=0 \end{equation} 2cos4х • cosх - cosх = 0
cosх • (2cos4х - 1) = 0
cosх = 0 2cos4х - 1 = 0
х = π/2 + kπ 2cos4х = 1
k ∈ Z cos4х = 1/2
4х = ±π/3 + 2kπ
х = ±-π/12 + πk/2
k ∈ Z;
✔ 2) cos2х = 1 + cos4х
cos2х - cos4х - 1 = 0
cos2х - (cos22х - sin22х) - (sin22х + cos22х) = 0
cos2х - cos22х + sin22х - sin22х - cos22х = 0
cos2х - 2cos22х = 0
cos2х(1 - 2cos2х) = 0
cos2х = 0 1 - 2cos2х = 0
2х = π/2 + kπ 2cos2х = 1
х = π/4 + kπ/2 cos2х = 1/2
k ∈ Z 2х = ±π/3 + 2kπ
х = ±π/6 + kπ, k ∈ Z.