Question

0,2^2cosx-1 - 26*0,2^cosx- 1/2 + 25=0

Answer 1

Правильный вариант условия: 0,2^(2cosx-1) - 26*0,2^(cosx- 1/2) + 25=0

$0,2^{2cosx-1}- 26*0,2^{cosx- \frac{1}{2} } + 25=0$
$0,2^{2(cosx-0,5)}- 26*0,2^{cosx- 0,5} + 25=0$
$(0,2^{(cosx-0,5)})^2- 26*0,2^{cosx- 0,5} + 25=0$
Замена:
$0,2^{cosx- 0,5} =a$
$a^2-26a+25=0$
$D=(-26)^2-4*1*25=576$
$a_1= \frac{26+24}{2}=25$
$a_2= \frac{26-24}{2}=1$
$0,2^{cosx- 0,5}=25$                или        $0,2^{cosx- 0,5}=1$
$5^{-(cosx- 0,5)}=5^2$                или        $0,2^{cosx- 0,5}=0,2^0$
${-cosx+ 0,5}=2$               или         $cosx-0,5=0$
$-cosx=1,5$                      или          $cosx=0,5$
$cosx=-1,5$                      или          $x=бarccos0,5+2 \pi n,$  $n$ ∈  $Z$
         ∅                                               $x=б \frac{ \pi }{3} +2 \pi n,$  $n$ ∈  $Z$

Answer 2

It looks like you have provided an equation involving trigonometric and exponential functions. However, there seems to be a formatting issue in your equation, which is making it a bit unclear. Let me try to interpret and solve the equation for you:

The equation you provided seems to be: $0.2^{2\cos(x) - 1} - 26 \cdot 0.2^{\cos(x)} - \frac{1}{2} + 25 = 0$

Let's simplify this equation step by step and solve for $x$:

Step 1: Combine the terms involving $0.2^{\cos(x)}$: $0.2^{2\cos(x) - 1} - 26 \cdot 0.2^{\cos(x)} - \frac{1}{2} + 25 = 0$ $0.2^{2\cos(x) - 1} - 26 \cdot 0.2^{\cos(x)} + \frac{49}{2} = 0$

Step 2: Let $y = 0.2^{\cos(x)}$. The equation becomes: $y^2 - 26y + \frac{49}{2} = 0$

Step 3: Solve the quadratic equation for $y$: $y^2 - 26y + \frac{49}{2} = 0$ $2y^2 - 52y + 49 = 0$ $2(y - 7)(y - 7) = 0$ $y = 7$

Step 4: Substitute $y$ back in terms of $0.2^{\cos(x)}$: $0.2^{\cos(x)} = 7$

Step 5: Solve for $\cos(x)$: $\cos(x) = \log_{0.2} 7$

Now, to find the value of $x$, you need to evaluate the inverse cosine (arccos) of $\log_{0.2} 7$. However, it's important to note that this solution might not have a simple closed-form expression and could require numerical methods to find an approximate value.

So, the solution for $x$ is: $x = \cos^{-1}(\log_{0.2} 7)$

Please note that this solution assumes there were no typos or errors in the original equation you provided. Double-check the equation for accuracy before proceeding with any calculations.