Question

Sin(5п-x)=cos(2x+7п)

Answer 1

2п из каждой функции просто убираем, они на них не влияют, получаем
Sin(п-х)=Cos(2х+п) 

теперь воспользуемся формулами приведения, получим
Sin(x)= - Cos (2x) 

Косинус переносим влево и применяем формулу двойного угла, получаем
Sin(x)+1-2Sin²(x)=0 (получили квадр уравнение, решаем через дискриминант)

-2Sin²x+Sinx+1=0   I*(-1))
2Sin²x-Sinx-1=0
D=1+4·2=9 , x1=-1, x2=2(не подх) 
Sin(x)=-1
х= - П/2+2Пn, n∈Z

Answer 2

I can help you with your math problem. Here is a detailed solution:

$$ \begin{aligned} \sin(5\pi-x) &= \cos(2x+7\pi) \\ \sin(5\pi-x) &= -\cos(7\pi-2x) & \text{(using the identity } \cos(\alpha+\beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \text{)} \\ \sin(x) &= \cos(2x) & \text{(using the periodicity of sine and cosine functions)} \\ \sin(x) &= \cos^2(x) - \sin^2(x) & \text{(using the identity } \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \text{)} \\ \sin(x) &= 1 - 2\sin^2(x) & \text{(using the identity } \sin^2(\theta) + \cos^2(\theta) = 1 \text{)} \\ 2\sin^2(x) + \sin(x) - 1 &= 0 & \text{(rearranging the terms)} \\ \end{aligned} $$

This is a quadratic equation in $\sin(x)$. To solve it, we can use the quadratic formula:

$$ \sin(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

where $a = 2$, $b = 1$, and $c = -1$. Plugging these values, we get:

$$ \sin(x) = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4} $$

Therefore, $\sin(x) = -1$ or $\sin(x) = \frac{1}{2}$. To find the values of $x$ that satisfy these equations, we can use the inverse sine function:

$$ x = \sin^{-1}(-1) \text{ or } x = \sin^{-1}\left(\frac{1}{2}\right) $$

The inverse sine function has a range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$. However, we can find other values of $x$ that are outside this range by using the periodicity of the sine function. The general solutions are:

$$ x = -\frac{\pi}{2} + 2\pi n \text{ or } x = \frac{\pi}{6} + 2\pi n \text{ or } x = \frac{5\pi}{6} + 2\pi n $$

where $n$ is any integer. These are the values of $x$ that make the original equation true.

I hope this helps you understand the problem. If you need more help, you can try [Mathway](https://www.mathway.com/trigonometry), a free online math solver. You can also learn more about trigonometric functions from [this article](https://www.krugosvet.ru/enc/matematika/trigonometricheskie-funkcii).

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