Question

4 sinx*cosx-x -3x=0

Answer 1

Ответ:

$\boldsymbol{x_1 = \frac{\pi}{4} + \pi n,n \in \Z} \\ \boldsymbol{x_2 = \arctan3 + \pi n,n \in \Z}$

Объяснение:

$4 \sin x \cdot \cos x - \sin {}^{2} x - 3 \cos {}^{2} x = 0$

Разделим оба части уравнения на cos²x , вспомним : sinx/cosx = tgx

$4 \tan x - \tan {}^{2} x - 3 = 0$

Сделаем замену tgx = t , тогда , tg²x = t²

$4t - t {}^{2} - 3 = 0 \\ t {}^{2} - 4t + 3 = 0 \\ D = ( - 4) {}^{2} - 4 \cdot3= 16 - 12 = 4 \\ \displaystyle t_1 = \frac{ - ( - 4) - \sqrt{4} }{8} = \frac{4 - 2}{2} = \frac{2}{2} = 1 \\ t_2 = \frac{ - ( - 4) + \sqrt{4} }{2} = \frac{4 + 2}{2} = \frac{6}{2} =3$

Вернемся к замене и распределим наши t :

$\tan x = 1 \\ x_1= \frac{\pi}{4} + \pi n,n \in \Z$

$\tan x = 3 \\ x_2= \arctan3 + \pi n,n \in \Z$

Answer 2

To solve the equation 4sin(x)cos(x) - x - 3x = 0, we can follow these steps:

1. Combine like terms: -x - 3x = -4x. The equation becomes: 4sin(x)cos(x) - 4x = 0.

2. Factor out 4: 4(sin(x)cos(x) - x) = 0.

3. Apply the zero-product property: Set each factor equal to zero and solve for x:

   Setting sin(x)cos(x) - x = 0: sin(x)cos(x) - x = 0.

   Setting 4 = 0 (which is not possible): 4 = 0.

Now, let's solve the equation sin(x)cos(x) - x = 0:

We can use the trigonometric identity sin(2x) = 2sin(x)cos(x) to simplify the equation:

sin(2x) - x = 0.

Now, we have a nonlinear equation. Unfortunately, there is no analytical solution for this equation. We need to use numerical methods or approximation techniques to find the solutions.