To identify the correct inverse trigonometric function to solve for the given angle, let's analyze each of the provided options carefully:
1. [tex]\(\tan^{-1}(0.71)\)[/tex]:
- The [tex]\(\tan^{-1}\)[/tex] function, or arctangent, can accept any real number as input and will yield an angle in the range [tex]\((- \frac{\pi}{2}, \frac{\pi}{2})\)[/tex]. Since 0.71 is a valid input for the arctangent function, this function is defined and valid.
2. [tex]\(\sin^{-1}(71)\)[/tex]:
- The [tex]\(\sin^{-1}\)[/tex] function, or arcsine, only accepts inputs in the range [tex]\([-1, 1]\)[/tex]. Since 71 is far outside this range, [tex]\(\sin^{-1}(71)\)[/tex] is undefined and invalid for finding an angle.
3. [tex]\(\cos^{-1}(1.41)\)[/tex]:
- The [tex]\(\cos^{-1}\)[/tex] function, or arccosine, also only accepts inputs in the range [tex]\([-1, 1]\)[/tex]. Since 1.41 is outside this range, [tex]\(\cos^{-1}(1.41)\)[/tex] is undefined and invalid for finding an angle.
4. [tex]\(\sin^{-1}(1.41)\)[/tex]:
- Similar to option (b), [tex]\(\sin^{-1}(1.41)\)[/tex] requires its input to be within the range [tex]\([-1, 1]\)[/tex]. Since 1.41 is outside this range, [tex]\(\sin^{-1}(1.41)\)[/tex] is undefined and invalid for finding an angle.
Given the analysis above, the only correct and valid inverse trigonometric function for solving for the given angle is:
[tex]\[
\tan^{-1}(0.71)
\][/tex]
Hence, the correct answer is:
[tex]\[
1
\][/tex]
