The number 𝑥
 lies in the range 0≤𝑥≤𝜋
 and satisfies cos(𝑥+24𝜋)=−3√2
.  What is 𝑥
?

The equation given is:

\[
\cos(x + 24\pi) = -\frac{3}{\sqrt{2}}
\]

### Step 1: Simplify the argument of the cosine function.

Since cosine is a periodic function with a period of \(2\pi\), adding \(24\pi\) to \(x\) does not change the value of \(x\) because:

\[
\cos(x + 24\pi) = \cos(x)
\]

Thus, the equation simplifies to:

\[
\cos(x) = -\frac{3}{\sqrt{2}}
\]

### Step 2: Analyze the equation.

The cosine function takes values between \(-1\) and \(1\), and \(-\frac{3}{\sqrt{2}} \approx -2.121\) is outside this range.

### Step 3: Conclusion.

Since \(-\frac{3}{\sqrt{2}}\) is not a valid value for \(\cos(x)\), there is no solution for \(x\) within the given range \(0 \leq x \leq \pi\). The problem seems to involve an impossible scenario for cosine.

### Questions for further reflection:
1. What is the range of values that the cosine function can take?
2. How does the periodicity of trigonometric functions affect their solutions?
3. What happens if the argument inside a trigonometric function exceeds its standard range?
4. How can we verify if a trigonometric equation has real solutions?
5. What strategies are helpful for solving more complex trigonometric equations?

#### Tip:
Always check if the output of a trigonometric function is within its valid range before proceeding to solve the equation.

The number x lies in the range 0 ≤ x ≤ π and satisfies cos(x + 24π) = −3/√2. What is x?

Explore how to solve the trigonometric equation cos(x + 24π) = −3/√2, focusing on cosine function properties and periodicity. Understand why the equation has no solution in the range 0 ≤ x ≤ π based on the valid range of cosine values. Suitable for students in Grades 11-12.

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