Question

what is the approximate value of x? round to the nearest tenth.
3.1 cm
3.9 cm
4.6 cm
5.4 cm

Explanation:

Step1: Identify the trigonometric ratio

Since it is a right - triangle, and we know the hypotenuse (\(6\) cm) and we want to find the side \(x\) opposite to the \(50^{\circ}\) angle. We use the sine ratio: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 50^{\circ}\), the hypotenuse \(c = 6\) cm, and the opposite side is \(x\). So, \(\sin(50^{\circ})=\frac{x}{6}\).

Step2: Solve for \(x\)

Multiply both sides of the equation \(\sin(50^{\circ})=\frac{x}{6}\) by \(6\). We get \(x = 6\times\sin(50^{\circ})\).
We know that \(\sin(50^{\circ})\approx0.766\). Then \(x=6\times0.766 = 4.596\approx4.6\) (rounded to the nearest tenth).

Answer:

\(4.6\) cm