Question

15
start
18.7
57°
x
16
13.4
10.5
14.2

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle with an angle of \(57^{\circ}\), hypotenuse \( = 16\), and we want to find the adjacent side \(x\) to the \(57^{\circ}\) angle. We use the cosine function, which is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Apply the cosine formula

We know that \(\cos(57^{\circ})=\frac{x}{16}\). To solve for \(x\), we can re - arrange the formula to \(x = 16\times\cos(57^{\circ})\).
We know that \(\cos(57^{\circ})\approx0.5446\) (using a calculator). Then \(x=16\times0.5446 = 8.7136\approx8.7\) (if we consider more precise calculation or rounding differences, but let's follow the steps). Wait, maybe I made a mistake in angle - side correspondence. Wait, the right angle, the angle of \(57^{\circ}\), so the side opposite to \(57^{\circ}\) and adjacent. Wait, maybe it's the sine function. Wait, no, let's re - check. If the angle is \(57^{\circ}\), and the hypotenuse is 16, and \(x\) is the side adjacent to \(57^{\circ}\), then \(\cos(57^{\circ})=\frac{x}{16}\), so \(x = 16\cos(57^{\circ})\). Let's calculate \(\cos(57^{\circ})\):
Using a calculator, \(\cos(57^{\circ})\approx0.5446\), so \(x = 16\times0.5446=8.7136\approx8.7\). But maybe the angle is such that we use sine. Wait, maybe the triangle is labeled differently. Wait, the right angle, the angle of \(57^{\circ}\), so if \(x\) is the opposite side, then \(\sin(57^{\circ})=\frac{x}{16}\), \(\sin(57^{\circ})\approx0.8387\), \(x = 16\times0.8387 = 13.4192\approx13.4\)? Wait, no, the number 13.4 is in the pentagon below. Wait, maybe I misread the triangle. Wait, the hypotenuse is 16, angle \(57^{\circ}\), and we want to find \(x\). Wait, maybe the side \(x\) is the opposite side. Let's re - examine the diagram. The right - angle is marked, the angle of \(57^{\circ}\), the hypotenuse is 16, and \(x\) is one of the legs. If we use \(\sin(57^{\circ})=\frac{x}{16}\), then \(x = 16\sin(57^{\circ})\). \(\sin(57^{\circ})\approx0.8387\), so \(x=16\times0.8387 = 13.4192\approx13.4\). Ah, that matches the number in the pentagon below (13.4). So I made a mistake in identifying adjacent and opposite. The angle of \(57^{\circ}\), the side \(x\) is opposite to the \(57^{\circ}\) angle, and the hypotenuse is 16. So we use \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin(57^{\circ})=\frac{x}{16}\), so \(x = 16\sin(57^{\circ})\).
Calculating \(\sin(57^{\circ})\approx0.8387\), then \(x = 16\times0.8387=13.4192\approx13.4\).

Answer:

\(x\approx13.4\)