Question

law of sines: $\frac{sin(a)}{a} = \frac{sin(b)}{b} = \frac{sin(c)}{c}$
triangle xyz with side xz = 2.6, angle at z is 76°, angle at y is 51°, side xy is z, side yz is x.
which equation is correct and can be used to solve for the value of z?

• $\frac{sin(51^circ)}{2.6} = \frac{sin(76^circ)}{z}$
• $\frac{sin(51^circ)}{2.6} = \frac{sin(53^circ)}{z}$
• $\frac{sin(76^circ)}{2.6} = \frac{sin(51^circ)}{z}$
• $\frac{sin(76^circ)}{2.6} = \frac{sin(53^circ)}{z}$

Explanation:

Step1: Find the measure of angle \(X\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(\angle X=180^{\circ}-(51^{\circ} + 76^{\circ})=180^{\circ}-127^{\circ}=53^{\circ}\)

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\). In \(\triangle XYZ\), if \(a = 2.6\) (side opposite \(\angle Y = 76^{\circ}\)) and \(c=z\) (side opposite \(\angle X = 53^{\circ}\)), then \(\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(53^{\circ})}{z}\)

Answer:

\(\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(53^{\circ})}{z}\) (the fourth option)