Question

Sin ( x/4 - 3 ) < -√2/2

Answer 1

Ответ:

) sin x = √2/2

 x = (-1)ⁿ × arcsin √2/2 + πn, n∈Z

 x = (-1)ⁿ × π/4 + πn, n∈Z

2) sin x = -√2/2

  x = (-1)ⁿ × arcsin (-√2/2) + πn, n∈Z

  x = (-1)ⁿ × -arcsin √2/2 + πn, n∈Z

  x = (-1)ⁿ × (-π/4) + πn, n∈Z

3) sin x = -√3/2

  x = (-1)ⁿ × arcsin (-√3/2) + πn, n∈Z

  x = (-1)ⁿ × -arcsin √3/2 + πn, n∈Z

  x = (-1)ⁿ × (-π/3) + πn, n∈Z

4) sin x = √3/2

  x = (-1)ⁿ × arcsin √3/2 + πn, n∈Z

  x = (-1)ⁿ × arcsin √3/2 + πn, n∈Z

  x = (-1)ⁿ × π/3 + πn, n∈Z

5) sin x = 4/5

  x = (-1)ⁿ × arcsin 4/5 + πn, n∈Z

  x = (-1)ⁿ × 0,927295 + πn, n∈Z

  x = (-1)ⁿ × 53,1° + πn, n∈Z

Пошаговое объяснение:

Answer 2

To solve the inequality $\sin\left(\frac{x}{4} - 3\right) < -\frac{\sqrt{2}}{2}$, we'll first find the solutions for $\frac{x}{4} - 3$. Then we'll determine the corresponding values of $x$ that satisfy the inequality for those solutions.

Let's start by finding the general solutions for $\frac{x}{4} - 3$ that satisfy the inequality $\sin\left(\frac{x}{4} - 3\right) < -\frac{\sqrt{2}}{2}$:

1. Find the general solutions for $\frac{x}{4} - 3$:

$$\frac{x}{4} - 3 = \arcsin\left(-\frac{\sqrt{2}}{2}\right) + 2k\pi$$

where $k$ is an integer.

Now, solve for $x$:

$$\frac{x}{4} = \arcsin\left(-\frac{\sqrt{2}}{2}\right) + 2k\pi + 3$$

Multiply both sides by 4 to isolate $x$:

$$x = 4\left(\arcsin\left(-\frac{\sqrt{2}}{2}\right) + 2k\pi + 3\right)$$

Now, we can substitute $k$ with different integers to find various solutions for $x$. However, we are interested in the solutions that satisfy the inequality $\sin\left(\frac{x}{4} - 3\right) < -\frac{\sqrt{2}}{2}$. For this, we'll consider the intervals where the sine function is negative and less than $-\frac{\sqrt{2}}{2}$.

The interval where $\sin\left(\frac{x}{4} - 3\right) < -\frac{\sqrt{2}}{2}$ is when $x$ is in the intervals $[4\arcsin(-\sqrt{2}/2) + 8k\pi, 4\arcsin(-\sqrt{2}/2) + 8k\pi + 2\pi]$ for $k$ being an integer.

Please note that this is a general solution, and you may want to substitute specific values for $k$ to find numerical solutions within a desired range.