To determine which equation correctly solves for [tex]\(\theta\)[/tex], we'll examine the mathematical operations and functions typically used to determine angles in trigonometry.
1. [tex]\(\theta = \sin^{-1}\left(\frac{4}{13}\right)\)[/tex]
2. [tex]\(\theta = \tan\left(\frac{4}{13}\right)\)[/tex]
3. [tex]\(\theta = \sin\left(\frac{4}{13}\right)\)[/tex]
4. [tex]\(\theta = \tan^{-1}\left(\frac{4}{13}\right)\)[/tex]
Step-by-step Explanation:
1. [tex]\(\theta = \sin^{-1}\left(\frac{4}{13}\right)\)[/tex]:
- The expression [tex]\(\sin^{-1}(x)\)[/tex] is the inverse sine function, often called arcsine. It solves for the angle [tex]\(\theta\)[/tex] such that [tex]\(\sin(\theta) = x\)[/tex].
- Here, [tex]\( \theta = \sin^{-1}\left(\frac{4}{13}\right) \)[/tex] would mean [tex]\(\sin(\theta) = \frac{4}{13}\)[/tex].
2. [tex]\(\theta = \tan\left(\frac{4}{13}\right)\)[/tex]:
- The expression [tex]\(\tan(x)\)[/tex] is the tangent function. It provides the ratio of the opposite side to the adjacent side for a given angle [tex]\(\theta\)[/tex].
- Here, [tex]\(\theta = \tan\left(\frac{4}{13}\right)\)[/tex] does not make sense because [tex]\(\tan\)[/tex] is not solving for [tex]\(\theta\)[/tex]; it's providing a ratio.
3. [tex]\(\theta = \sin\left(\frac{4}{13}\right)\)[/tex]:
- The expression [tex]\(\sin(x)\)[/tex] is the sine function. It calculates the ratio of the opposite side to the hypotenuse for a given angle [tex]\(\theta\)[/tex].
- Here, [tex]\(\theta = \sin\left(\frac{4}{13}\right)\)[/tex] does not make sense because [tex]\(\sin\)[/tex] is not solving for [tex]\(\theta\)[/tex]; it's calculating a ratio.
4. [tex]\(\theta = \tan^{-1}\left(\frac{4}{13}\right)\)[/tex]:
- The expression [tex]\(\tan^{-1}(x)\)[/tex] is the inverse tangent function, often called arctangent. It solves for the angle [tex]\(\theta\)[/tex] such that [tex]\(\tan(\theta) = x\)[/tex].
- Here, [tex]\(\theta = \tan^{-1}\left(\frac{4}{13}\right) \)[/tex] would mean [tex]\(\tan(\theta) = \frac{4}{13}\)[/tex].
Therefore, the equation that correctly solves for [tex]\(\theta\)[/tex] in this context is:
[tex]\[
\theta = \tan^{-1}\left(\frac{4}{13}\right)
\][/tex]
The correct choice is:
[tex]\[
\boxed{\theta = \tan^{-1}\left(\frac{4}{13}\right)}
\][/tex]
