Question

Sinx + sin|x| = 0 .​

Answer 1

Ответ:

$x\in(-\infty;\;0)\cup\{n\pi\},n\in\mathbb{Z},\;n\ge0$

Объяснение:

$sinx+sin|x|=0$

Раскроем модуль:

$sinx+sinx=0,\;x\ge0\\sinx-sinx=0,\;x$

Во втором случае получаем 0=0 => Любой x, который меньше 0 подходит.

В 1-ом случае:

$sinx=0\\x=n\pi,\;n\in\mathbb{Z}$

Тогда ответом будет:

$x=n\pi,\;n\in\mathbb{Z},\;n\ge0\\x$

Заметьте, что корень x=0 мы учли в 1-ом случае, так как 0*\pi=0.

Answer 2

The equation you've provided is $\sin(x) + \sin(|x|) = 0$.

Let's analyze the solutions to this equation. First, we can consider the behavior of the two terms separately:

1. $\sin(x)$ is a periodic function with a period of $2\pi$. It oscillates between -1 and 1. So, it crosses the x-axis infinitely many times within each period.

2. $\sin(|x|)$ reflects the positive and negative parts of the sine curve across the y-axis, creating a symmetric shape. As the absolute value $|x|$ ensures that the input to the sine function is always positive, this portion of the curve is also restricted to the range [-1, 1] and has infinite intersections with the x-axis.

Now, let's consider the combined equation $\sin(x) + \sin(|x|) = 0$. Since both terms are constrained within the range [-1, 1], the sum of the two terms can be zero in the following cases:

1. When $\sin(x) = 0$, which occurs at $x = k\pi$, where $k$ is an integer.

2. When $\sin(|x|) = 0$, which occurs at $|x| = n\pi$, where $n$ is a non-negative integer. This gives $x = n\pi$ or $x = -n\pi$.

So, the solutions to the equation $\sin(x) + \sin(|x|) = 0$ are given by:

1. $x = k\pi$ where $k$ is an integer.
2. $x = n\pi$ where $n$ is a non-negative integer.
3. $x = -n\pi$ where $n$ is a non-negative integer.

These solutions represent the points at which the combined curve $\sin(x) + \sin(|x|)$ intersects the x-axis and the sum of the two sine terms equals zero.