Question

which is the best approximation of the value of x? use the law of sines to find the answer. 1.5 units 2.1 units law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

1. First, find the third - angle of the triangle:

• The sum of the interior angles of a triangle is \(180^{\circ}\). Given two angles \(A = 75^{\circ}\) and \(B = 50^{\circ}\), the third - angle \(C=180-(75 + 50)=55^{\circ}\).
• By the law of sines, \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Let the side opposite the \(75^{\circ}\) angle be \(a = 2\), the side opposite the \(50^{\circ}\) angle be \(b=x\), and the side opposite the \(55^{\circ}\) angle be \(c\).
• Using the law of sines \(\frac{\sin A}{a}=\frac{\sin B}{b}\), we substitute \(A = 75^{\circ}\), \(a = 2\), and \(B = 50^{\circ}\), \(b=x\).
• So, \(\frac{\sin75^{\circ}}{2}=\frac{\sin50^{\circ}}{x}\).
• We know that \(\sin75^{\circ}=\sin(45^{\circ}+30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx\frac{2.449 + 1.414}{4}=\frac{3.863}{4}=0.966\), and \(\sin50^{\circ}\approx0.766\).
• Then, from \(\frac{\sin75^{\circ}}{2}=\frac{\sin50^{\circ}}{x}\), we can cross - multiply to get \(x\times\sin75^{\circ}=2\times\sin50^{\circ}\).
• \(x=\frac{2\times\sin50^{\circ}}{\sin75^{\circ}}\).
• Substituting the approximate values of \(\sin50^{\circ}\) and \(\sin75^{\circ}\), we have \(x=\frac{2\times0.766}{0.966}\approx1.58\approx1.6\). The closest value to \(1.6\) among the options is \(1.5\) units.

Answer:

1.5 units