Question

find the length of side x in simplest radical form with a rational denominator.
answer attempt 1 out of 2
x =

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\sin60^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the side opposite to \(60^{\circ}\) is \(5\) and the hypotenuse is \(x\). So, \(\sin60^{\circ}=\frac{5}{x}\).

Step2: Substitute the value of \(\sin60^{\circ}\)

Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{5}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(x\times\sqrt{3}=2\times5\), so \(x = \frac{10}{\sqrt{3}}\).

Step4: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{3}\): \(x=\frac{10\sqrt{3}}{3}\).

Answer:

\(\frac{10\sqrt{3}}{3}\)