Let's solve the inequality [tex]\( 2 \sin x \geq \sqrt{2} \)[/tex] step-by-step.
### Step 1: Isolate the sine function
First, we want to isolate [tex]\(\sin x\)[/tex] on one side of the inequality.
[tex]\[ 2 \sin x \geq \sqrt{2} \][/tex]
To do this, we divide both sides of the inequality by 2:
[tex]\[ \sin x \geq \frac{\sqrt{2}}{2} \][/tex]
### Step 2: Evaluate [tex]\(\frac{\sqrt{2}}{2}\)[/tex]
Next, let's find the value of [tex]\(\frac{\sqrt{2}}{2}\)[/tex]:
[tex]\[ \frac{\sqrt{2}}{2} = 0.7071067811865476 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
We need to determine the values of [tex]\(x\)[/tex] for which [tex]\(\sin x \geq 0.7071067811865476\)[/tex]. The solution for [tex]\(\sin x = 0.7071067811865476\)[/tex] is:
[tex]\[ x = \arcsin \left(0.7071067811865476\right) \][/tex]
This gives us:
[tex]\[ x = 0.7853981633974484 \][/tex]
However, the sine function [tex]\(\sin x\)[/tex] is periodic and symmetric. The sine function reaches the value [tex]\(0.7071067811865476\)[/tex] again in the interval [tex]\([0, 2\pi]\)[/tex]:
[tex]\[ x = \pi - \arcsin(0.7071067811865476) \][/tex]
This gives us:
[tex]\[ x = \pi - 0.7853981633974484 = 2.356194490192345 \][/tex]
### Step 4: Determine the Interval for [tex]\(\sin x \geq 0.7071067811865476\)[/tex]
The sine function is increasing on the interval [tex]\([0, \frac{\pi}{2}]\)[/tex] and decreasing on the interval [tex]\([\frac{\pi}{2}, \pi]\)[/tex]. Therefore, the inequality [tex]\(\sin x \geq 0.7071067811865476\)[/tex] holds for:
[tex]\[ x \in \left[0.7853981633974484, 2.356194490192345\right] \][/tex]
### Summary
The solution to [tex]\(2 \sin x \geq \sqrt{2}\)[/tex] is:
- The sine function value which is calculated as [tex]\(\frac{\sqrt{2}}{2} = 0.7071067811865476\)[/tex]
- The interval where [tex]\(x\)[/tex] lies is [tex]\(x \in \left[ 0.7853981633974484, 2.356194490192345 \right] \)[/tex]
Thus, the detailed step-by-step solution concludes that the values [tex]\(x\)[/tex] should lie within the interval:
[tex]\[ \left[0.7853981633974484, 2.356194490192345\right] \][/tex]
