Let's identify the appropriate inverse trigonometric function for each of the given expressions and determine their corresponding angles.
a. [tex]\(\tan^{-1}(3.83)\)[/tex]
- The function [tex]\(\tan^{-1}(x)\)[/tex] (also written as [tex]\(\arctan(x)\)[/tex]) returns the angle whose tangent is [tex]\(x\)[/tex]. The value of [tex]\(\arctan(3.83)\)[/tex] is approximately [tex]\(1.315401373737063\)[/tex] radians.
b. [tex]\(\sin^{-1}(3.83)\)[/tex]
- The function [tex]\(\sin^{-1}(x)\)[/tex] (also written as [tex]\(\arcsin(x)\)[/tex]) returns the angle whose sine is [tex]\(x\)[/tex]. However, the sine function only has an inverse for values in the range [tex]\([-1, 1]\)[/tex]. Since [tex]\(3.83\)[/tex] is outside this range, [tex]\(\sin^{-1}(3.83)\)[/tex] is undefined and cannot be calculated. Hence, it is represented as `None`.
c. [tex]\(\cos^{-1}(0.26)\)[/tex]
- The function [tex]\(\cos^{-1}(x)\)[/tex] (also written as [tex]\(\arccos(x)\)[/tex]) returns the angle whose cosine is [tex]\(x\)[/tex]. Since [tex]\(0.26\)[/tex] is within the allowable range [tex]\([-1, 1]\)[/tex], the value of [tex]\(\arccos(0.26)\)[/tex] is approximately [tex]\(1.3077741238864278\)[/tex] radians.
d. [tex]\(\sin^{-1}(0.26)\)[/tex]
- The function [tex]\(\sin^{-1}(x)\)[/tex] (also written as [tex]\(\arcsin(x)\)[/tex]) returns the angle whose sine is [tex]\(x\)[/tex]. Since [tex]\(0.26\)[/tex] is within the allowable range [tex]\([-1, 1]\)[/tex], the value of [tex]\(\arcsin(0.26)\)[/tex] is approximately [tex]\(0.2630222029084689\)[/tex] radians.
### Summary of Results:
1. [tex]\(\tan^{-1}(3.83)\)[/tex] ≈ [tex]\(1.315401373737063\)[/tex] radians
2. [tex]\(\sin^{-1}(3.83)\)[/tex] = Not defined (None)
3. [tex]\(\cos^{-1}(0.26)\)[/tex] ≈ [tex]\(1.3077741238864278\)[/tex] radians
4. [tex]\(\sin^{-1}(0.26)\)[/tex] ≈ [tex]\(0.2630222029084689\)[/tex] radians
