Rationale
Tan x = 5,000/7,000 is a correct way to calculate x.
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. So, if we know the lengths of the opposite and adjacent sides, we can calculate the angle using the tangent function.
A) sin x=5,000 /7,000
This equation would be appropriate if we were dealing with the ratio of the length of the opposite side to the hypotenuse in a right triangle. However, the question does not make it clear if we are given the lengths of the opposite side and hypotenuse. Therefore, this equation is not necessarily correct.
B) sin x=7,000 /5,000
Similarly to option A, the sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. But this equation suggests a value greater than 1 for sin x, which is not possible as the sine of an angle can never exceed 1. Hence, this equation is incorrect.
C) tan x=5,000 /7,000
As stated, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. This equation correctly uses the tangent function to calculate the angle, x, given the lengths of the opposite and adjacent sides.
D) tan x=7,000 /5,000
This equation uses the tangent function, but it reverses the ratio of the side lengths. If the length of the opposite side is 5,000 and the length of the adjacent side is 7,000, then the ratio should be 5,000/7,000, not 7,000/5,000. Hence, this equation is incorrect.
Conclusion
The correct way to calculate the angle x in a right triangle, given the lengths of the opposite and adjacent sides, is with the equation tan x = 5,000/7,000. The other options either use the incorrect function (sin instead of tan) or incorrectly reverse the ratio of the side lengths.
