Question

triangle abc is a right triangle and sin(53°) = 4/x. solve for x and round to the nearest whole number. which equation correctly uses the value of x to represent the cosine of angle a?

Explanation:

Step1: Recall sine definition

In right - triangle \(ABC\), \(\sin(A)=\frac{BC}{AB}\). Given \(\sin(53^{\circ})=\frac{4}{x}\), and we know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).
Since \(\sin(53^{\circ})=\frac{4}{x}\), we can cross - multiply: \(x\times\sin(53^{\circ}) = 4\).
We know that \(\sin(53^{\circ})\approx0.7986\), so \(x=\frac{4}{\sin(53^{\circ})}\approx\frac{4}{0.7986}\approx5\).

Step2: Recall cosine definition

In right - triangle \(ABC\), \(\cos(A)=\frac{AC}{AB}\). Here, the adjacent side to angle \(A\) is \(y\) and the hypotenuse is \(x\).
We know that \(\sin(53^{\circ})=\frac{4}{x}\), and by the Pythagorean theorem \(y=\sqrt{x^{2}-4^{2}}\). Also, \(\cos(53^{\circ})=\frac{y}{x}\).
Since we found \(x\approx5\), and \(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}\), for angle \(A = 53^{\circ}\), \(\cos(53^{\circ})=\frac{y}{x}\).

Answer:

The value of \(x\approx5\) and the correct equation for \(\cos(53^{\circ})\) is \(\cos(53^{\circ})=\frac{y}{x}\). Among the given options, if \(x\approx5\), the correct equation for \(\cos(53^{\circ})\) is \(\cos(53^{\circ})=\frac{y}{x}\), but if we assume the options are based on the sides given in the problem - setup, and since \(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}\), when \(\sin(53^{\circ})=\frac{4}{x}\), then \(\cos(53^{\circ})=\frac{y}{x}\). If we consider the relationship between sides, and knowing \(\sin(53^{\circ})=\frac{4}{x}\), by Pythagorean theorem \(y=\sqrt{x^{2}-16}\), and \(\cos(53^{\circ})=\frac{y}{x}\). The first option \(\cos(53^{\circ})=\frac{y}{x}\) (assuming the first option \(\cos(53^{\circ})=\frac{y}{x}\) is what is meant by the first box in the multiple - choice part as it is the correct cosine formula for angle \(A\) in right - triangle \(ABC\)).