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 (1)
The problem requires finding the triangle where cos ( x ) = 6.7 4.3 .
Analyze each triangle to check if the ratio of the adjacent side to the hypotenuse matches the given condition.
Both triangle A and D satisfy the condition.
Choose triangle A as the answer. A
Explanation
Understanding the problem The problem states that we need to find the triangle in which x = cos − 1 ( 6.7 4.3 ) . This means we are looking for a triangle where cos ( x ) = 6.7 4.3 . In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse.
Analyzing the triangles Let's examine each triangle to see if the ratio of the adjacent side to the hypotenuse is equal to 6.7 4.3 .
Triangle A: cos ( x ) = 6.7 4.3 . This matches the given condition. Triangle B: cos ( x ) = 4.3 6.7 . This does not match the given condition. Triangle C: cos ( x ) = 4.3 6.7 . This does not match the given condition. Triangle D: cos ( x ) = 6.7 4.3 . This matches the given condition.
Identifying the correct triangle Both triangle A and triangle D satisfy the condition cos ( x ) = 6.7 4.3 . However, we need to choose the correct triangle based on the diagram. Since the images may not be drawn to scale, we should look for the triangle where 4.3 is the adjacent side and 6.7 is the hypotenuse. Both triangles A and D have this configuration.
Final Answer Since both triangles A and D have the correct ratio of adjacent to hypotenuse, and the problem does not provide any other distinguishing information, we can choose either A or D. However, without additional information or a more precise diagram, it's impossible to definitively choose one over the other. Assuming there is only one correct answer, we should re-examine the triangles. In triangle A, 4.3 is adjacent to angle x and 6.7 is the hypotenuse. In triangle D, 4.3 is adjacent to angle x and 6.7 is the hypotenuse. Therefore, both triangles fit the description.
Conclusion Since both triangles A and D satisfy the condition, and the problem statement does not give any other information to differentiate them, we can assume that either A or D would be considered correct. However, let's choose triangle A.
Examples
Understanding trigonometric ratios like cosine is crucial in various real-world applications, such as calculating the angles and distances in construction, navigation, and surveying. For instance, if you're building a ramp and know the desired height (opposite side) and the length of the ramp (hypotenuse), you can use the sine function to find the angle of elevation. Similarly, in navigation, the cosine function helps determine the distance and direction to a destination based on angles and known distances.
