Question

Cos(3x)*cos(6x)=cos(4x)*cos(7x) на промежутке [Pi/3;Pi/2]

Answer 1

$\cos(3x) \cos(6x) = \cos(4x) \cos(7x) \\ \frac{ \cos(3x) }{2} + \frac{ \cos(9x) }{2} = \frac{ \cos(3x) }{2} + \frac{ \cos(11x) }{2} \\ \cos(9x) = \cos(11x) \\ \cos(9x) - \cos(11x) = 0 \\ 2 \sin(x) \sin(10x) = 0 \\ \sin(x) = 0 \\ \sin(10x) = 0 \\ x = \pi n \\ 10x = \pi m \\ x = \pi n, \: n \in Z \\ x = \frac{\pi}{10} m, \: m \in Z$

При m = 10 корни совпадают поэтому мы оставляем меньший из них:

$x = \frac{\pi}{10} l, \: l \in Z$

Отберем корни на даном отрезке:

$\frac{\pi}{3} \leqslant \frac{\pi}{10} l \leqslant \frac{\pi}{2} \\ \frac{1}{3} \leqslant \frac{l}{10} \leqslant \frac{1}{2} \\ \frac{10}{3} \leqslant l \leqslant 5 , \: l \in Z \\ l = 4: \: x = \frac{4\pi}{10} = \frac{2\pi}{5} \\ l = 5: \: x = \frac{5\pi}{10} = \frac{\pi}{2}$

Answer 2

To solve the equation cos(3x) * cos(6x) = cos(4x) * cos(7x) on the interval [π/3, π/2], we can use trigonometric identities to simplify the equation and then solve for x.

First, let's rewrite the equation using the cosine addition formula: cos(3x) * [cos(6x) + cos(4x)] = 0

Using the identity cos(A) + cos(B) = 2 * cos((A + B) / 2) * cos((A - B) / 2), we can rewrite the equation as: cos(3x) * [2 * cos((6x + 4x) / 2) * cos((6x - 4x) / 2)] = 0

Simplifying further: cos(3x) * [2 * cos(5x) * cos(x)] = 0

Now, we have two factors: cos(3x) = 0 and cos(5x) * cos(x) = 0.

Let's solve them one by one:

1. cos(3x) = 0: To find the values of x for which cos(3x) = 0, we need to solve the equation: 3x = π/2 + k * π (where k is an integer)

Solving for x: x = (π/2 + k * π) / 3 (where k is an integer)

Since we are interested in the interval [π/3, π/2], let's find the values of k that satisfy this interval:

For k = 0: x = π/6

For k = 1: x = (π/2 + π) / 3 = π/6 + π/3 = π/2

Therefore, the values of x for cos(3x) = 0 in the given interval are x = π/6 and x = π/2.

2. cos(5x) * cos(x) = 0: To find the values of x for which cos(5x) * cos(x) = 0, we need to solve the equations: cos(5x) = 0 or cos(x) = 0

For cos(5x) = 0: 5x = π/2 + k * π (where k is an integer)

Solving for x: x = (π/2 + k * π) / 5 (where k is an integer)

For cos(x) = 0: x = π/2 + k * π/2 (where k is an integer)

Again, let's find the values of k that satisfy the interval [π/3, π/2]:

For cos(5x) = 0: For k = 1: x = (π/2 + π) / 5 = π/10 + π/5 = 3π/10

For k = 2: x = (π/2 + 2π) / 5 = π/10 + 2π/5 = 7π/10

For k = 3: x = (π/2 + 3π) / 5 = π/10 + 3π/5 = 11π/10 (outside the given interval)

For cos(x) = 0: For k = 1: x = π/2 + π/2 = π

Therefore, the values of x for cos(5x) * cos(x) = 0 in the given interval are x = π/10, 3π/10, and π.

In summary, the solutions for the equation cos(3x) * cos(6x) = cos(4x) * cos(7x) on the interval [π/3, π/2] are: x = π/6, π/2, π/10, 3π/10, and π.