Question

Log2(2-cosx)=1+2log2(-sinx)

Answer 1

Loq2 (2 - cosx )=Loq2 2 +2Loq2 (-sinx) ,  ОДЗ : sinx < 0  
2 -cosx =2sin²x ;
2 -cosx  =2(1-cos²x) ;
2cosx(cosx -1/2) =0 ;
[ cosx =0 ;cosx= 1/2,
cosx=0⇒sinx =+ -1⇒sinx = -1⇔x = -π/2 +2π*k ,k∈Z.
cosx= 1/2⇒x = -π/3+2π*k  ,k∈Z.

ответ : π/2 +2π*k     ;   -π/3+2π*k  ,k∈Z.

Answer 2

To solve the equation $\log_2(2 - \cos(x)) = 1 + 2\log_2(-\sin(x))$, we'll first simplify it and then solve for $x$.

1. Start with the original equation:

$\log_2(2 - \cos(x)) = 1 + 2\log_2(-\sin(x))$

2. Apply logarithmic properties:

$\log_2(2 - \cos(x)) = \log_2(2) + \log_2((-\sin(x))^2)$

3. Combine the logarithmic terms on the right side:

$\log_2(2 - \cos(x)) = \log_2(2) + \log_2(\sin^2(x))$

4. Use the properties of logarithms to combine the two logarithmic terms on the right side:

$\log_2(2 - \cos(x)) = \log_2(2 \cdot \sin^2(x))$

5. Remove the logarithm by setting the expressions inside the logarithms equal to each other:

$2 - \cos(x) = 2 \cdot \sin^2(x)$

6. Now, solve for $x$. First, subtract $2 \cdot \sin^2(x)$ from both sides:

$-\cos(x) = 2 \cdot \sin^2(x) - 2$

7. Next, use the identity $\sin^2(x) + \cos^2(x) = 1$ to replace $\sin^2(x)$ in the equation:

$-\cos(x) = 2(1 - \cos^2(x)) - 2$

8. Distribute and simplify:

$-\cos(x) = 2 - 2\cos^2(x) - 2$

9. Combine like terms:

$-\cos(x) = -2\cos^2(x)$

10. Divide both sides by $-1$ to make the equation easier to work with:

$\cos(x) = 2\cos^2(x)$

11. Rearrange the equation:

$2\cos^2(x) - \cos(x) = 0$

12. Factor out $\cos(x)$:

$\cos(x)(2\cos(x) - 1) = 0$

Now, set each factor equal to zero and solve for $x$:

1. $\cos(x) = 0$

2. $2\cos(x) - 1 = 0$

Let's solve each of these equations:

1. For $\cos(x) = 0$, you get $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

2. For $2\cos(x) - 1 = 0$, you get $2\cos(x) = 1$, and then $\cos(x) = \frac{1}{2}$. This corresponds to $x = \frac{\pi}{3} + 2n\pi$ and $x = \frac{5\pi}{3} + 2n\pi$, where $n$ is an integer.

So, the solutions for $x$ are $x = \frac{\pi}{2} + n\pi$, $x = \frac{\pi}{3} + 2n\pi$, and $x = \frac{5\pi}{3} + 2n\pi$, where $n$ can be any integer.