Question

Решите уравнение: Cos(x - П/2) = -1

Answer 1

$\cos(x - \frac{\pi}{2} ) = - 1 \\ \sin(x) = - 1 \\ x = \frac{3\pi}{2} + 2k\pi$

Answer 2

$\cos(x - \frac{\pi}{2} ) = - 1 \\ \cos( - \frac{\pi}{2} + x ) = - 1 \\ \cos( - ( \frac{\pi}{2} - x ) = - 1 \\ \cos( \frac{\pi}{2} - x ) = - 1 \\ \sin(x ) = - 1 \\ x = \frac{3\pi}{2} + 2k\pi$

Answer 3

To solve the equation $\cos(x - \frac{\pi}{2}) = -1$, we need to find the values of $x$ that satisfy the equation.

First, let's recall the value of the cosine function at certain angles:

$\cos(0) = 1$ $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ $\cos(\frac{\pi}{3}) = \frac{1}{2}$ $\cos(\frac{\pi}{2}) = 0$ $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$ $\cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}$ $\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$ $\cos(\pi) = -1$

Now, we want to solve the equation $\cos(x - \frac{\pi}{2}) = -1$. This means that $x - \frac{\pi}{2}$ should be an odd multiple of $\pi$ (i.e., $\pi, 3\pi, 5\pi, \ldots$) since the cosine function is equal to -1 at odd multiples of $\pi$.

So, we set up the equation:

$x - \frac{\pi}{2} = (2n + 1)\pi$

where $n$ is an integer representing the number of complete cycles of cosine that have been completed before reaching the value -1.

Now, solve for $x$:

$x = (2n + 1)\pi + \frac{\pi}{2}$

So, the solution to the equation is:

$x = \left(2n + \frac{3}{2}\right)\pi$

where $n$ is an integer. This represents an infinite set of solutions because there are infinitely many odd multiples of $\pi$. For each value of $n$, we get a unique solution for $x$.