Domain of sin^-1[1+coax]+(√49-x^2) (where [] denotes greatest integer function)

To determine the domain of the expression sin^-1[1 + cos(x)] + √(49 - x²), we need to analyze each component carefully. The domain will be defined by the restrictions imposed by both the inverse sine function and the square root function.

Understanding the Components

Let's break down the expression into its parts:

• sin^-1[1 + cos(x)]: The inverse sine function, sin^-1(y), is defined for values of y in the range of [-1, 1]. Therefore, we need to ensure that the argument [1 + cos(x)] falls within this range.
• √(49 - x²): The square root function is defined for non-negative values. Thus, we require that 49 - x² ≥ 0.

Finding the Domain of Each Component

Let's start with the first part:

1. Analyzing sin^-1[1 + cos(x)]

For the expression [1 + cos(x)] to be valid in the inverse sine function, we need:

• -1 ≤ 1 + cos(x) ≤ 1

This can be simplified into two inequalities:

• 1 + cos(x) ≥ -1 → cos(x) ≥ -2 (always true since cos(x) ranges from -1 to 1)
• 1 + cos(x) ≤ 1 → cos(x) ≤ 0

The second inequality, cos(x) ≤ 0, means that x must be in the intervals where the cosine function is non-positive. This occurs in the intervals:

• π/2 + 2kπ ≤ x ≤ 3π/2 + 2kπ, where k is any integer.

2. Analyzing √(49 - x²)

Next, we look at the square root function:

• 49 - x² ≥ 0

This simplifies to:

• x² ≤ 49
• -7 ≤ x ≤ 7

Combining the Results

Now, we need to find the intersection of the two sets of conditions we derived:

• From sin^-1[1 + cos(x)], we have the intervals where cos(x) ≤ 0.
• From √(49 - x²), we have the interval [-7, 7].

Thus, the valid x-values must satisfy both conditions. The intervals where cos(x) is non-positive within the range of [-7, 7] can be determined by evaluating the cosine function at specific points:

• For x = π/2, cos(π/2) = 0 (valid)
• For x = 3π/2, cos(3π/2) = 0 (valid)

Since π/2 is approximately 1.57 and 3π/2 is approximately 4.71, both points fall within the interval [-7, 7]. Therefore, the intervals where cos(x) is non-positive within the range of [-7, 7] are:

• From π/2 to 3π/2, which is approximately [1.57, 4.71].

Final Domain

In conclusion, the domain of the expression sin^-1[1 + cos(x)] + √(49 - x²) is:

• x ∈ [π/2, 3π/2] ∩ [-7, 7]

Thus, the final domain is approximately:

• x ∈ [π/2, 3π/2] = [1.57, 4.71]

This means that for values of x within this interval, the entire expression is defined and valid.