Identify the correct inverse trigonometric function to use to solve for the given angle.
a. [tex]$\tan ^{-1}(3.83)$[/tex]
b. [tex]$\sin ^{-1}(3.83)$[/tex]
c. [tex]$\cos ^{-1}(.26)$[/tex]
d. [tex]$\sin ^{-1}(.26)$[/tex]
Answer (2)
The domain of sin − 1 ( x ) and cos − 1 ( x ) is [ − 1 , 1 ] .
The domain of tan − 1 ( x ) is ( − ∞ , ∞ ) .
Check each option against the domain restrictions.
Options a, c, and d are valid, but without more context, we cannot determine a single correct answer. Option b is definitively incorrect because 3.83 is not in the domain of the inverse sine function.
Possible correct answers are a, c, and d. a , c , d
Explanation
Analyze the domains of inverse trigonometric functions. We need to identify the correct inverse trigonometric function to use for solving an angle. Let's analyze the given options and their domains.
The domain of the inverse sine function, denoted as sin − 1 ( x ) or arcsin ( x ) , is − 1 ≤ x ≤ 1 .
The domain of the inverse cosine function, denoted as cos − 1 ( x ) or arccos ( x ) , is − 1 ≤ x ≤ 1 .
The domain of the inverse tangent function, denoted as tan − 1 ( x ) or arctan ( x ) , is all real numbers, i.e., ( − ∞ , ∞ ) .
Check each option against the domain restrictions. Now, let's examine each option:
a. tan − 1 ( 3.83 ) : Since the domain of tan − 1 ( x ) is all real numbers, 3.83 is a valid input. Thus, this option is potentially correct.
b. sin − 1 ( 3.83 ) : Since the domain of sin − 1 ( x ) is − 1 ≤ x ≤ 1 , and 1"> 3.83 > 1 , this is not a valid option.
c. cos − 1 ( 0.26 ) : Since the domain of cos − 1 ( x ) is − 1 ≤ x ≤ 1 , and − 1 ≤ 0.26 ≤ 1 , this is a valid option.
d. sin − 1 ( 0.26 ) : Since the domain of sin − 1 ( x ) is − 1 ≤ x ≤ 1 , and − 1 ≤ 0.26 ≤ 1 , this is a valid option.
Determine the correct inverse trigonometric function. Based on the domain restrictions, options a, c, and d are valid. However, the question asks to identify the correct inverse trigonometric function. Without additional information, such as the ratio of sides in a right triangle, we cannot definitively choose between options a, c, and d. However, the question does not provide any context, so any of the valid options could be used depending on the information available. The options b is the only one that is definitely incorrect.
Final Analysis Since options a, c, and d are all potentially correct depending on the context, and the question asks to identify the correct inverse trigonometric function, we must consider the information given. The question only provides the value inside the inverse trigonometric function. Therefore, any of the options a, c, or d could be correct. However, option b is definitively incorrect because 3.83 is not in the domain of the inverse sine function.
Without more context, it is impossible to determine a single correct answer. However, if we assume the question is asking which function could be used, and we are given a ratio of sides, then we can determine the correct function. If the ratio is opposite/adjacent, we use tan − 1 . If the ratio is adjacent/hypotenuse, we use cos − 1 . If the ratio is opposite/hypotenuse, we use sin − 1 .
Conclusion Based on the analysis, options a, c, and d are valid, but without more context, we cannot determine a single correct answer. However, since the question asks to identify the correct inverse trigonometric function, and we have no additional information, we can assume that any of the valid options could be correct. Option b is definitively incorrect.
Therefore, the possible correct answers are: a. tan − 1 ( 3.83 ) c. cos − 1 ( .26 ) d. sin − 1 ( .26 )
Since the question asks for the correct inverse trigonometric function, and multiple options are valid, we must assume there is an error in the question. However, if we assume the question is asking which of the options could be used, then any of the valid options are correct. If we assume the question is asking which option is always valid, then the answer is a. tan − 1 ( 3.83 ) , since the domain of tan − 1 ( x ) is all real numbers.
However, without further context, we cannot definitively say which option is the correct answer. We can only say that option b is incorrect.
Examples
Inverse trigonometric functions are used in various real-world applications, such as calculating angles in construction, navigation, and physics. For example, if you know the lengths of the sides of a right triangle, you can use the arctangent function ( tan − 1 ) to find the angle of elevation of a ramp or the angle between two walls in a building. Similarly, the arcsine ( sin − 1 ) and arccosine ( cos − 1 ) functions are used in GPS systems to determine the latitude and longitude of a location based on the signals received from satellites. These functions are also crucial in computer graphics for rendering 3D objects on a 2D screen, ensuring that objects appear correctly from different viewpoints.
The correct inverse trigonometric functions options are a ( tan − 1 ( 3.83 ) ), c ( cos − 1 ( 0.26 ) ), and d ( sin − 1 ( 0.26 ) ). Option b is incorrect because 3.83 is outside the domain for sin − 1 ( x ) . Therefore, multiple options may be correct, depending on the specific problem.
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