Question

what is the approximate value of k? use the law of sines to find the answer. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$ 2.9 units 3.8 units 5.1 units 6.2 units

Explanation:

Step1: Find angle J

The sum of angles in a triangle is 180°. So, $\angle J=180^{\circ}-120^{\circ}- 40^{\circ}=20^{\circ}$.

Step2: Apply the law of sines

According to the law of sines $\frac{\sin(J)}{2}=\frac{\sin(L)}{k}$. We know $\sin(J)=\sin(20^{\circ})$, $\sin(L)=\sin(40^{\circ})$ and the side opposite $\angle J$ is 2. Substituting the values, we get $\frac{\sin(20^{\circ})}{2}=\frac{\sin(40^{\circ})}{k}$.

Step3: Solve for k

Cross - multiply: $k\times\sin(20^{\circ})=2\times\sin(40^{\circ})$. Then $k = \frac{2\times\sin(40^{\circ})}{\sin(20^{\circ})}$. Since $\sin(40^{\circ}) = 2\sin(20^{\circ})\cos(20^{\circ})$, $k=\frac{2\times2\sin(20^{\circ})\cos(20^{\circ})}{\sin(20^{\circ})}=4\cos(20^{\circ})$. Using $\cos(20^{\circ})\approx0.9397$, $k\approx4\times0.9397 = 3.7588\approx3.8$.

Answer:

3.8 units