Identify the correct inverse trigonometric function to use to solve for the given angle.
a. [tex]$\tan ^{-1}(.71)$[/tex]
b. [tex]$\sin ^{-1}(.71)$[/tex]
c. [tex]$\cos ^{-1}(1.41)$[/tex]
d. [tex]$\sin ^{-1}(1.41)$[/tex]
Answer (2)
The domain of sin − 1 ( x ) and cos − 1 ( x ) is − 1 ≤ x ≤ 1 .
The domain of tan − 1 ( x ) is all real numbers.
Check each option to see if the value is within the domain of the inverse trigonometric function.
Options a and b are valid, while options c and d are invalid. a , b
Explanation
Problem Analysis We need to identify the correct inverse trigonometric function to solve for an angle. Let's analyze the given options and their domains.
Domain of Inverse Trigonometric Functions The domain of the inverse sine function, sin − 1 ( x ) , and the inverse cosine function, cos − 1 ( x ) , is − 1 ≤ x ≤ 1 . This means that the value inside the inverse sine or inverse cosine must be between -1 and 1, inclusive. The domain of the inverse tangent function, tan − 1 ( x ) , is all real numbers.
Checking the Options Now, let's examine each option:
a. tan − 1 ( 0.71 ) : Since 0.71 is a real number, this option is valid because the domain of tan − 1 ( x ) is all real numbers.
b. sin − 1 ( 0.71 ) : Since − 1 ≤ 0.71 ≤ 1 , this option is also valid because 0.71 falls within the domain of sin − 1 ( x ) .
c. cos − 1 ( 1.41 ) : Since 1.41 > 1, this option is invalid because 1.41 is outside the domain of cos − 1 ( x ) .
d. sin − 1 ( 1.41 ) : Since 1.41 > 1, this option is invalid because 1.41 is outside the domain of sin − 1 ( x ) .
Conclusion Both options a and b are valid inverse trigonometric functions. Without additional information, we cannot determine which one is the 'correct' one. However, since the question asks to identify the correct inverse trigonometric function, and both a and b are valid, we can assume that the question is flawed. However, based on the domain restrictions, options c and d are definitely incorrect.
Examples
Imagine you are building a ramp and need to determine the angle of elevation. If you know the opposite and adjacent sides of the right triangle formed by the ramp, you would use the arctangent ( tan − 1 ) function to find the angle. Similarly, if you know the opposite side and the hypotenuse, you would use the arcsine ( sin − 1 ) function. The arccosine ( cos − 1 ) function is used when you know the adjacent side and the hypotenuse. These inverse trigonometric functions are essential tools in various fields like engineering, physics, and navigation to calculate angles based on side ratios.
The valid inverse trigonometric functions to solve for an angle from the given options are (a) tan − 1 ( 0.71 ) and (b) sin − 1 ( 0.71 ) . Options (c) and (d) are invalid as their inputs exceed the allowed range. Therefore, the choices are (a) and (b).
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