Question

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

Answer 1

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:

Answer 2

To solve the trigonometric equation $\cos\left(\frac{\pi}{2}+5x\right)+\sin(x)=2\cos(3x)$, we'll use various trigonometric identities to simplify and solve the equation step by step.

First, let's manipulate the equation using trigonometric identities:

1. Use angle addition identity for cosine: $\cos\left(\frac{\pi}{2}+5x\right) = -\sin(5x)$

2. Use angle addition identity for sine: $\sin(x) = \sin\left(\frac{\pi}{2}-\left(\frac{\pi}{2}-x\right)\right) = \sin\left(\frac{\pi}{2}-y\right) = \cos(y)$ where $y = \frac{\pi}{2}-x$.

The equation becomes: $-\sin(5x) + \cos(y) = 2\cos(3x)$

3. Use angle addition identity for cosine: $\cos(y) = \cos\left(\frac{\pi}{2}-\left(\frac{\pi}{2}-x\right)\right) = \sin\left(\frac{\pi}{2}-x\right)$

The equation becomes: $-\sin(5x) + \sin\left(\frac{\pi}{2}-x\right) = 2\cos(3x)$

4. Use angle subtraction identity for sine: $\sin\left(\frac{\pi}{2}-x\right) = \cos(x)$

The equation becomes: $-\sin(5x) + \cos(x) = 2\cos(3x)$

5. Use triple angle identity for cosine: $\cos(3x) = 4\cos^3(x) - 3\cos(x)$

Substituting this into the equation: $-\sin(5x) + \cos(x) = 2(4\cos^3(x) - 3\cos(x))$

Now, this equation is in terms of $x$. You can solve for $x$ using numerical methods or graphing software since it doesn't have a simple algebraic solution. Numerical methods like the Newton-Raphson method or graphing the functions on both sides of the equation and finding the intersection points can help you find approximate solutions.