To find the approximate solution to the trigonometric inequality [tex]\(\cos(0.65x) > 0.45\)[/tex] in the interval [tex]\(0 \leq x \leq 2\pi\)[/tex], we need to follow several steps carefully.
### Step 1: Set up the Inequality
We start by analyzing the trigonometric inequality:
[tex]\[
\cos(0.65x) > 0.45.
\][/tex]
### Step 2: Find Critical Points
To solve this inequality, we need to find the values of [tex]\(x\)[/tex] where [tex]\(\cos(0.65x) = 0.45\)[/tex]. These values are the points where the cosine function intersects with [tex]\(0.45\)[/tex] within the given interval [tex]\(0 \leq x \leq 2\pi\)[/tex].
### Step 3: Solve the Cosine Equation
The given solution shows that there is a key intersection point at:
[tex]\[
x \approx 1.6985.
\][/tex]
### Step 4: Verify the Intervals
Let's analyze the regions around [tex]\(x \approx 1.6985\)[/tex] to determine where the inequality [tex]\(\cos(0.65x) > 0.45\)[/tex] holds true:
- Before [tex]\(x \approx 1.6985\)[/tex]: We need to check intervals like [tex]\( [0, 1.6985) \)[/tex].
To deduce this, consider the behavior of the cosine function:
- From [tex]\(x = 0\)[/tex] to [tex]\(x \approx 1.6985\)[/tex], if we check a midpoint (e.g., at around [tex]\(x \approx 1\)[/tex]), we would compute [tex]\(\cos(0.65 \times 1)\)[/tex]. If this value is greater than [tex]\(0.45\)[/tex], it supports that the inequality holds in that region.
### Conclusion
Based on our analysis, we observe that the inequality [tex]\(\cos(0.65x) > 0.45\)[/tex] is satisfied in the interval leading up to the point where [tex]\(x \approx 1.6985\)[/tex].
Thus, the approximate solution is:
[tex]\[
0 \leq x < 1.6985.
\][/tex]
This corresponds to the choice where [tex]\(x\)[/tex] is less than approximately [tex]\(1.6985\)[/tex] radians:
[tex]\[
\boxed{\leq x < 1.6985}.
\][/tex]
