In which triangle is the value of $x$ equal to $\cos ^{-1}\left(\frac{4.3}{6.7}\right)$? (Images may not be drawn to scale.)
Answer (2)
The problem states that x = cos − 1 ( 6.7 4.3 ) , implying cos ( x ) = 6.7 4.3 .
We need to find the triangle where the ratio of the adjacent side to the hypotenuse for angle x is 6.7 4.3 .
The correct triangle is the one where the adjacent side is 4.3 and the hypotenuse is 6.7.
Therefore, the triangle where x = cos − 1 ( 6.7 4.3 ) is the triangle where the ratio of the adjacent side to the hypotenuse for angle x is 6.7 4.3 .
Explanation
Understanding the problem We are given that x = cos − 1 ( 6.7 4.3 ) . This means that cos ( x ) = 6.7 4.3 . We need to identify the triangle in which the cosine of angle x is equal to 6.7 4.3 . In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse.
Identifying the correct triangle We are looking for a triangle where the side adjacent to angle x has length 4.3 and the hypotenuse has length 6.7. Without the images of the triangles, we can only describe the properties of the correct triangle.
Conclusion Therefore, the triangle in which x = cos − 1 ( 6.7 4.3 ) is the triangle where the ratio of the adjacent side to the hypotenuse for angle x is 6.7 4.3 .
Examples
Imagine you are building a ramp and need to determine the angle of elevation. If the horizontal distance (adjacent side) is 4.3 meters and the length of the ramp (hypotenuse) is 6.7 meters, you can use the inverse cosine function to find the angle. This ensures the ramp has the correct slope for its intended use, whether it's for accessibility or recreational purposes.
The value of x = cos − 1 ( 6.7 4.3 ) indicates that it is the angle in a right triangle where the adjacent side is 4.3 and the hypotenuse is 6.7. The ratio of these sides gives us the cosine of the angle. Thus, we find a triangle with these dimensions to understand the angle's value.
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