Предмет: Математика, автор: vladsaltan

cos(pi/2+5x)+sinx=2cos3x

Ответы

Автор ответа: vladmusienkoa

Ответ:

Воспользуемся формулой приведения для косинуса.

Из уравнения cos (пи/2 + 5x) + sin x = 2 * cos 3x получим равносильное уравнение:

-sin 5x + sin x = 2 * cos 3x, что в соответствии с формулой разности синусов равносильно:

2 * sin ((x - 5x) / 2) * cos ((x + 5x) / 2) = 2 * cos 3x.

Отсюда: sin (-2x) * cos (3x) = cos 3x, то есть cos 3x * (1 + sin 2x) = 0.

Тогда cos 3x = 0 или sin 2x = -1.

В первом случае 3x = пи/2 + 2 * пи * n, где n - целое. То есть:

x = пи/6 + 2/3 * пи * n, n ∈ Z.

Во втором случае 2x = -пи/2 + 2 * пи * k, где k - целое. То есть:

x = -пи/4 + пи * k, k ∈ Z.

Ответ: x1 = пи/6 + 2/3 * пи * n, n ∈ Z; x2 = -пи/4 + пи * k, k ∈ Z.

Пошаговое объяснение:

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