Rationale
tan x = 5,000 / 7,000.
The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, applies here. Therefore, the correct way to calculate x when given the two side lengths is using the tangent function, which leads us to option C.
A) sin x = 5,000 / 7,000
This equation incorrectly uses the sine function, which represents the ratio of the opposite side to the hypotenuse, not the adjacent side. In a right triangle, sine is not applicable for calculating x when given the lengths of the opposite and adjacent sides.
B) sin x = 7,000 / 5,000
Similar to option A, this choice also misapplies the sine function. The sine function cannot be used directly with the lengths of the opposite and adjacent sides; hence it does not correctly represent the relationship needed to solve for x.
C) tan x = 5,000 / 7,000
This is the correct equation as it properly uses the tangent function, which is defined as the ratio of the lengths of the opposite side (5,000) to the adjacent side (7,000). This formulation accurately allows for the calculation of angle x in a right triangle.
D) tan x = 7,000 / 5,000
This choice incorrectly states the tangent ratio, reversing the relationship between the opposite and adjacent sides. For the correct calculation of x, the opposite side must be in the numerator, making this option incorrect.
Conclusion
To calculate angle x accurately when given the lengths of two sides of a right triangle, the tangent function must be employed with the correct ratio of opposite over adjacent. Option C correctly applies this principle, while the other options either misuse the sine function or present the tangent ratio incorrectly. Understanding these relationships is crucial for solving trigonometric equations effectively.
