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 8.2 units

Explanation:

Step1: Find the third - angle

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

Step2: Apply the law of sines

The law of sines is $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$. We know that $\frac{\sin\angle K}{k}=\frac{\sin\angle L}{l}$. Here, $\angle K = 120^{\circ}$, $\angle L=40^{\circ}$, and $l = 2$. So $\frac{\sin120^{\circ}}{k}=\frac{\sin40^{\circ}}{2}$.

Step3: Solve for k

Cross - multiply to get $k\times\sin40^{\circ}=2\times\sin120^{\circ}$. Then $k=\frac{2\times\sin120^{\circ}}{\sin40^{\circ}}$. Since $\sin120^{\circ}=\frac{\sqrt{3}}{2}\approx0.866$ and $\sin40^{\circ}\approx0.643$, $k=\frac{2\times0.866}{0.643}=\frac{1.732}{0.643}\approx2.7$. The closest value to 2.7 among the options is 2.9 units.

Answer:

2.9 units