Question

Cos (3x+π/4)=корень2/2

Answer 1

$\cos(3x + \frac{\pi}{4} ) = \frac{ \sqrt{2} }{2} \\ 3x + \frac{\pi}{4} = \pm \frac{\pi}{4} + 2\pi n \\ \left[ \begin{gathered} 3x = \frac{\pi}{4} - \frac{\pi}{4} + 2\pi n \\ 3x = -\frac{\pi}{4}-\frac{\pi}{4}+2\pi n \end{gathered} \right. \\ \left[ \begin{gathered} x = \frac{2\pi}{3} n \\ x = -\frac{\pi}{6}+ \frac{2\pi}{3} n, \: n \in Z \end{gathered} \right.$

Answer 2

To solve the equation $\cos(3x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, you can follow these steps:

Step 1: Use the inverse cosine function to find the angle whose cosine is $\frac{\sqrt{2}}{2}$. Since the cosine of $45^\circ$ (or $\frac{\pi}{4}$) is $\frac{\sqrt{2}}{2}$, you know that one possible solution to the equation is $3x + \frac{\pi}{4} = \frac{\pi}{4}$, which simplifies to $3x = 0$. So, $x = 0$.

Step 2: Since the cosine function is periodic with a period of $2\pi$, you can add integer multiples of $2\pi$ to the solution. This is because the cosine function repeats its values every $2\pi$ radians.

So, the general solution for $x$ is: $x = \frac{0}{3} + \frac{2\pi \cdot n}{3},$ where $n$ is an integer.

This gives you an infinite set of solutions, where $x$ can take any value of the form $\frac{2\pi \cdot n}{3}$, where $n$ is an integer.

Keep in mind that this is just one possible way to solve the equation. There might be additional solutions or alternative methods to approach this problem.