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 measure of angle \(X\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(\angle X=180^{\circ}-75^{\circ}-50^{\circ}=55^{\circ}\)

Step2: Apply the Law of Sines

By the Law of Sines, \(\frac{\sin(Y)}{y}=\frac{\sin(Z)}{z}\). Here, \(z = 2\), \(\angle Y = 75^{\circ}\), \(\angle Z=50^{\circ}\)
So, \(\frac{\sin(75^{\circ})}{y}=\frac{\sin(50^{\circ})}{2}\)
Cross - multiply: \(y\times\sin(50^{\circ})=2\times\sin(75^{\circ})\)
\(y=\frac{2\times\sin(75^{\circ})}{\sin(50^{\circ})}\)
We know that \(\sin(75^{\circ})=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966\), \(\sin(50^{\circ})\approx0.766\)
\(y=\frac{2\times0.966}{0.766}\approx2.5\)

Answer:

2.5 units