Question

use the law of sines to find the value of y. round to the nearest tenth. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$ 1.4 units 1.6 units 2.5 units 2.6 units

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is 180°. So the third - angle $\angle X=180^{\circ}-75^{\circ}-50^{\circ}=55^{\circ}$.

Step2: Apply the law of sines

According to the law of sines $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$. Here, we can use $\frac{\sin(Y)}{y}=\frac{\sin(Z)}{z}$. We know $z = 2$, $\angle Y=75^{\circ}$, $\angle Z = 50^{\circ}$. So $\frac{\sin(75^{\circ})}{y}=\frac{\sin(50^{\circ})}{2}$.

Step3: Solve for y

Cross - multiply to get $y\times\sin(50^{\circ})=2\times\sin(75^{\circ})$. Then $y=\frac{2\times\sin(75^{\circ})}{\sin(50^{\circ})}$. Since $\sin(75^{\circ})=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966$ and $\sin(50^{\circ})\approx0.766$. Then $y=\frac{2\times0.966}{0.766}\approx2.5$.

Answer:

2.5 units