Question

solve for x.
15 cm
35°
x
x = ? cm
round to the nearest hundredth.

Explanation:

Step1: Identify trigonometric ratio

In a right - triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For the given right - triangle, the hypotenuse is \(15\) cm, the angle is \(35^{\circ}\), and the side opposite to the \(35^{\circ}\) angle is \(x\). So, we use the sine function: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), where \(\theta = 35^{\circ}\), opposite \(=x\), and hypotenuse \( = 15\). So, \(\sin(35^{\circ})=\frac{x}{15}\).

Step2: Solve for x

To solve for \(x\), we can multiply both sides of the equation \(\sin(35^{\circ})=\frac{x}{15}\) by \(15\). So, \(x = 15\times\sin(35^{\circ})\).

We know that \(\sin(35^{\circ})\approx0.5736\) (using a calculator). Then \(x=15\times0.5736 = 8.604\). Rounding to the nearest hundredth, we look at the thousandth place. Since the digit in the thousandth place is \(4\), which is less than \(5\), we round down. So, \(x\approx8.60\).

Answer:

\(8.60\)