Question

which expression can be used to find the value of x?
○ \\(\frac{(\sin 32^{\circ})(\sin 47^{\circ})}{11}\\)
○ \\(11(\sin 32^{\circ})(\sin 47^{\circ})\\)
○ \\(\frac{11(\sin 47^{\circ})}{\sin 32^{\circ}}\\)
○ \\(\frac{11(\sin 32^{\circ})}{\sin 47^{\circ}}\\)

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where $a, b, c$ are the lengths of the sides opposite angles $A, B, C$ respectively.

Step2: Identify the sides and angles in triangle \(JKL\)

In triangle \(JKL\), side \(JL = 11\) is opposite angle \(K = 47^\circ\), and side \(JK=x\) is opposite angle \(L = 32^\circ\).

Step3: Apply the Law of Sines

Using the Law of Sines, we have $\frac{x}{\sin 32^\circ}=\frac{11}{\sin 47^\circ}$.

Step4: Solve for \(x\)

Cross - multiplying gives \(x=\frac{11\sin 32^\circ}{\sin 47^\circ}\).

Answer:

\(\frac{11(\sin 32^{\circ})}{\sin 47^{\circ}}\) (the fourth option)