Answer :
To determine which inverse trigonometric functions are valid for solving the given angle, we need to analyze each option carefully based on the properties and domains of these functions. Let's go through each option one by one:
### Option a: [tex]\(\tan^{-1}(3.83)\)[/tex]
The function [tex]\(\tan^{-1}(x)\)[/tex] gives us an angle whose tangent is [tex]\(x\)[/tex]. The tangent function, [tex]\(\tan(\theta)\)[/tex], can take any real number as its value, so the domain of [tex]\(\tan^{-1}(x)\)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex]. In this case, [tex]\(\tan^{-1}(3.83)\)[/tex] is valid because [tex]\(3.83\)[/tex] is within this domain. Specifically, the angle [tex]\(\theta\)[/tex] for [tex]\(\tan^{-1}(3.83)\)[/tex] is approximately [tex]\(1.3154\)[/tex] radians.
### Option b: [tex]\(\sin^{-1}(3.83)\)[/tex]
The function [tex]\(\sin^{-1}(x)\)[/tex] gives us an angle whose sine is [tex]\(x\)[/tex]. The sine function, [tex]\(\sin(\theta)\)[/tex], ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. Therefore, the domain of [tex]\(\sin^{-1}(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. Since [tex]\(3.83\)[/tex] is outside of this range, [tex]\(\sin^{-1}(3.83)\)[/tex] is not valid. There is no real angle whose sine is [tex]\(3.83\)[/tex].
### Option c: [tex]\(\cos^{-1}(0.26)\)[/tex]
The function [tex]\(\cos^{-1}(x)\)[/tex] gives us an angle whose cosine is [tex]\(x\)[/tex]. The cosine function, [tex]\(\cos(\theta)\)[/tex], also ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. The domain of [tex]\(\cos^{-1}(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. Here, [tex]\(0.26\)[/tex] is within the valid range of the cosine function. Hence, [tex]\(\cos^{-1}(0.26)\)[/tex] is valid. The angle [tex]\(\theta\)[/tex] for [tex]\(\cos^{-1}(0.26)\)[/tex] is approximately [tex]\(1.3078\)[/tex] radians.
### Option d: [tex]\(\sin^{-1}(0.26)\)[/tex]
The function [tex]\(\sin^{-1}(x)\)[/tex] gives us an angle whose sine is [tex]\(x\)[/tex]. As previously noted, the sine function ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. The domain of [tex]\(\sin^{-1}(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. Since [tex]\(0.26\)[/tex] falls within this range, [tex]\(\sin^{-1}(0.26)\)[/tex] is valid. The angle [tex]\(\theta\)[/tex] for [tex]\(\sin^{-1}(0.26)\)[/tex] is approximately [tex]\(0.2630\)[/tex] radians.
### Summary
After analyzing each option, we come to the following conclusions:
- [tex]\(\tan^{-1}(3.83)\)[/tex] is valid with an angle approximately [tex]\(1.3154\)[/tex] radians.
- [tex]\(\sin^{-1}(3.83)\)[/tex] is not valid.
- [tex]\(\cos^{-1}(0.26)\)[/tex] is valid with an angle approximately [tex]\(1.3078\)[/tex] radians.
- [tex]\(\sin^{-1}(0.26)\)[/tex] is valid with an angle approximately [tex]\(0.2630\)[/tex] radians.
Therefore, the correct inverse trigonometric functions to solve for the given angle are:
- [tex]\(\tan^{-1}(3.83)\)[/tex]
- [tex]\(\cos^{-1}(0.26)\)[/tex]
- [tex]\(\sin^{-1}(0.26)\)[/tex]
Invalid function:
- [tex]\(\sin^{-1}(3.83)\)[/tex]
### Option a: [tex]\(\tan^{-1}(3.83)\)[/tex]
The function [tex]\(\tan^{-1}(x)\)[/tex] gives us an angle whose tangent is [tex]\(x\)[/tex]. The tangent function, [tex]\(\tan(\theta)\)[/tex], can take any real number as its value, so the domain of [tex]\(\tan^{-1}(x)\)[/tex] is all real numbers [tex]\((-\infty, \infty)\)[/tex]. In this case, [tex]\(\tan^{-1}(3.83)\)[/tex] is valid because [tex]\(3.83\)[/tex] is within this domain. Specifically, the angle [tex]\(\theta\)[/tex] for [tex]\(\tan^{-1}(3.83)\)[/tex] is approximately [tex]\(1.3154\)[/tex] radians.
### Option b: [tex]\(\sin^{-1}(3.83)\)[/tex]
The function [tex]\(\sin^{-1}(x)\)[/tex] gives us an angle whose sine is [tex]\(x\)[/tex]. The sine function, [tex]\(\sin(\theta)\)[/tex], ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. Therefore, the domain of [tex]\(\sin^{-1}(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. Since [tex]\(3.83\)[/tex] is outside of this range, [tex]\(\sin^{-1}(3.83)\)[/tex] is not valid. There is no real angle whose sine is [tex]\(3.83\)[/tex].
### Option c: [tex]\(\cos^{-1}(0.26)\)[/tex]
The function [tex]\(\cos^{-1}(x)\)[/tex] gives us an angle whose cosine is [tex]\(x\)[/tex]. The cosine function, [tex]\(\cos(\theta)\)[/tex], also ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. The domain of [tex]\(\cos^{-1}(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. Here, [tex]\(0.26\)[/tex] is within the valid range of the cosine function. Hence, [tex]\(\cos^{-1}(0.26)\)[/tex] is valid. The angle [tex]\(\theta\)[/tex] for [tex]\(\cos^{-1}(0.26)\)[/tex] is approximately [tex]\(1.3078\)[/tex] radians.
### Option d: [tex]\(\sin^{-1}(0.26)\)[/tex]
The function [tex]\(\sin^{-1}(x)\)[/tex] gives us an angle whose sine is [tex]\(x\)[/tex]. As previously noted, the sine function ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. The domain of [tex]\(\sin^{-1}(x)\)[/tex] is [tex]\([-1, 1]\)[/tex]. Since [tex]\(0.26\)[/tex] falls within this range, [tex]\(\sin^{-1}(0.26)\)[/tex] is valid. The angle [tex]\(\theta\)[/tex] for [tex]\(\sin^{-1}(0.26)\)[/tex] is approximately [tex]\(0.2630\)[/tex] radians.
### Summary
After analyzing each option, we come to the following conclusions:
- [tex]\(\tan^{-1}(3.83)\)[/tex] is valid with an angle approximately [tex]\(1.3154\)[/tex] radians.
- [tex]\(\sin^{-1}(3.83)\)[/tex] is not valid.
- [tex]\(\cos^{-1}(0.26)\)[/tex] is valid with an angle approximately [tex]\(1.3078\)[/tex] radians.
- [tex]\(\sin^{-1}(0.26)\)[/tex] is valid with an angle approximately [tex]\(0.2630\)[/tex] radians.
Therefore, the correct inverse trigonometric functions to solve for the given angle are:
- [tex]\(\tan^{-1}(3.83)\)[/tex]
- [tex]\(\cos^{-1}(0.26)\)[/tex]
- [tex]\(\sin^{-1}(0.26)\)[/tex]
Invalid function:
- [tex]\(\sin^{-1}(3.83)\)[/tex]
