principal value of Sin-1 (√3-1/2√2). From inverse trigonometry.please give solution

To find the principal value of sin^-1(√3 - 1/2√2), we need to first clarify what we're working with. The principal value of the inverse sine function, sin^-1(x), is defined for x in the range of -1 to 1, and it specifically gives us an angle in the range of -π/2 to π/2. Let's break this down step by step to find the solution.

Step 1: Evaluate the Expression

We start with the expression √3 - 1/2√2. To make this easier to process, let's calculate its numerical value:

• √3 is approximately 1.732.
• 1/2√2 simplifies to approximately 1/2 * 1.414 = 0.707.
• Now, calculate: 1.732 - 0.707 = 1.025.

This means we're actually looking for sin^-1(1.025). However, this value is outside the range of the sin^-1 function since its domain is limited to [-1, 1]. Thus, we need to check if the original expression can be simplified or represented in a different way.

Step 2: Check Values Within Domain

Since 1.025 is outside the valid range for sin^-1(x), we conclude that sin^-1(√3 - 1/2√2) does not produce a valid angle. However, let’s verify what happens when we consider the limits of the functions:

• If we had a value that’s less than or equal to 1, we could find the angle.
• For example, if we had sin^-1(0.5), we know that the corresponding angle would be π/6 or 30 degrees.

In contrast, because √3 - 1/2√2 exceeds 1, we cannot find a corresponding angle in the range of sin^-1.

Step 3: Consider Possible Adjustments

If you wanted to explore the function further, you could look for angles where the sine values are known. For example, values like sin(0), sin(π/6), or sin(π/4) are easily accessible. However, since we’ve established that √3 - 1/2√2 is greater than 1, no such sin^-1 angle exists.

Final Thoughts

In summary, the principal value of sin^-1(√3 - 1/2√2) does not yield a valid answer due to the input being outside the domain of the arcsin function. If you encounter similar expressions, always check to see if they fall within the acceptable range to ensure a valid result.