Which expressions can be used to find -m? Select two options.
[tex]\cos ^{-1}\left(\frac{9.8}{6.3}\right)[/tex]
[tex]\cos ^{-1}\left(\frac{6.3}{9.8}\right)[/tex]
[tex]\cos ^{-1}\left(\frac{7.5}{9.8}\right)[/tex]
[tex]\sin ^{-1}\left(\frac{9.8}{7.5}\right)[/tex]
[tex]\sin ^{-1}\left(\frac{7.5}{9.8}\right)[/tex]
Answer (2)
The expressions that can be used to find -m are \text{cos}^{-1}igg(\frac{6.3}{9.8}igg) and \text{sin}^{-1}igg(\frac{7.5}{9.8}igg). Both expressions are defined and fall within the appropriate domain for inverse trigonometric functions.
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Check the domain of inverse trigonometric functions to eliminate undefined expressions.
Assume a right triangle with sides 6.3, 7.5, and hypotenuse 9.8.
Express angles α and β using inverse sine and cosine functions.
The expressions that can be used to find -m are cos − 1 ( 9.8 6.3 ) and sin − 1 ( 9.8 7.5 ) .
Explanation
Understanding the Problem We are given five expressions involving inverse trigonometric functions. Our goal is to identify two expressions that could represent − m , where m is an angle. We need to consider the domains of inverse trigonometric functions. The arguments of cos − 1 ( x ) and sin − 1 ( x ) must be between -1 and 1, inclusive, i.e., − 1 ≤ x ≤ 1 .
Checking the Domain Let's examine each option:
cos − 1 ( 6.3 9.8 ) : Since 1"> 6.3 9.8 ≈ 1.55 > 1 , this expression is not defined.
cos − 1 ( 9.8 6.3 ) : Since 9.8 6.3 ≈ 0.64 < 1 , this expression is defined.
cos − 1 ( 9.8 7.5 ) : Since 9.8 7.5 ≈ 0.77 < 1 , this expression is defined.
sin − 1 ( 7.5 9.8 ) : Since 1"> 7.5 9.8 ≈ 1.31 > 1 , this expression is not defined.
sin − 1 ( 9.8 7.5 ) : Since 9.8 7.5 ≈ 0.77 < 1 , this expression is defined.
Finding Possible Expressions Now, let's assume that there is a right triangle with sides 6.3 , 7.5 , and hypotenuse 9.8 . Let α be the angle opposite the side 7.5 , and β be the angle opposite the side 6.3 . Then, we have:
sin ( α ) = 9.8 7.5 ⇒ α = sin − 1 ( 9.8 7.5 ) cos ( α ) = 9.8 6.3 ⇒ α = cos − 1 ( 9.8 6.3 ) sin ( β ) = 9.8 6.3 ⇒ β = sin − 1 ( 9.8 6.3 ) cos ( β ) = 9.8 7.5 ⇒ β = cos − 1 ( 9.8 7.5 )
If − m = α , then the possible expressions are sin − 1 ( 9.8 7.5 ) and cos − 1 ( 9.8 6.3 ) .
If − m = β , then the possible expressions are sin − 1 ( 9.8 6.3 ) and cos − 1 ( 9.8 7.5 ) .
Final Answer From the given options, the two expressions that are defined and can be used to find − m are:
cos − 1 ( 9.8 6.3 ) and sin − 1 ( 9.8 7.5 ) .
Examples
Imagine you're designing a roof for a house. The slope of the roof can be described using angles. If you know the lengths of the sides of the triangle formed by the roof, you can use inverse trigonometric functions to find the angles. This helps ensure the roof is built at the correct angle for water runoff and structural stability. Similarly, in navigation, if you know the distance to an object and its height, you can use inverse trigonometric functions to find the angle of elevation, which is crucial for determining the object's location.
