Question

(tg x + 2) * cos x = 0

Answer 1

Tg(x)=sin(x)/cos(x);
Значит, что (tg(x)+2)*cos(x)= 0
Cos(x)<>0, tg(x)= -2;
X = arctg(-2)+Pik, k- целые числа

Answer 2

To solve the equation $(tg x + 2) \cdot \cos x = 0$, we need to find the values of $x$ that satisfy the equation. To do this, we'll follow these steps:

Step 1: Identify the values that make $\tan x + 2$ or $\cos x$ equal to zero. Step 2: Find the solutions for $x$ based on the identified values.

Step 1: Identifying the values that make $\tan x + 2$ or $\cos x$ equal to zero:

The equation can only be zero if either $\tan x + 2 = 0$ or $\cos x = 0$.

a) $\tan x + 2 = 0$: To find the values of $x$ that satisfy this, we'll solve for $x$ by subtracting 2 from both sides and then taking the tangent inverse (arctan):

$\tan x = -2$

$x = \arctan(-2) + k\pi$, where $k$ is an integer.

b) $\cos x = 0$: The cosine function is zero at odd multiples of $\frac{\pi}{2}$. So we have:

$x = \frac{\pi}{2} + k\pi$ or $x = \frac{3\pi}{2} + k\pi$, where $k$ is an integer.

Step 2: Find the solutions for $x$ based on the identified values:

The solutions for $x$ are the values that satisfy either $\tan x + 2 = 0$ or $\cos x = 0$:

1. From $\tan x + 2 = 0$: $x = \arctan(-2) + k\pi$, where $k$ is an integer.
2. From $\cos x = 0$: $x = \frac{\pi}{2} + k\pi$ or $x = \frac{3\pi}{2} + k\pi$, where $k$ is an integer.

To summarize, the general solutions to the equation $(tg x + 2) \cdot \cos x = 0$ are:

1. $x = \arctan(-2) + k\pi$, where $k$ is an integer.
2. $x = \frac{\pi}{2} + k\pi$, where $k$ is an integer.
3. $x = \frac{3\pi}{2} + k\pi$, where $k$ is an integer.