Question

law of sines
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
(diagram: triangle xyz with ( xy = 2 ), ( angle y = 75^circ ), ( angle z = 50^circ ), side ( xz = y ))

Explanation:

Step1: Find angle at X

In triangle \( XYZ \), the sum of angles in a triangle is \( 180^\circ \). So, \( \angle X=180^\circ - 75^\circ - 50^\circ = 55^\circ \).

Step2: Apply Law of Sines

Law of Sines: \( \frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c} \). Here, side opposite \( \angle X \) is \( YZ = 2 \), side opposite \( \angle Z \) is \( XY = y \). So, \( \frac{\sin(55^\circ)}{2}=\frac{\sin(50^\circ)}{y} \).

Step3: Solve for y

Cross - multiply: \( y\times\sin(55^\circ)=2\times\sin(50^\circ) \). Then \( y = \frac{2\times\sin(50^\circ)}{\sin(55^\circ)} \). Calculate \( \sin(50^\circ)\approx0.7660 \), \( \sin(55^\circ)\approx0.8192 \). So, \( y=\frac{2\times0.7660}{0.8192}=\frac{1.532}{0.8192}\approx1.9 \)? Wait, no, wait. Wait, angle at Z is \( 50^\circ \), side opposite angle Z is XY (length y), angle at X is \( 55^\circ \), side opposite angle X is YZ (length 2). Wait, maybe I mixed up. Wait, angle at Y is \( 75^\circ \), side opposite angle Y is XZ? Wait, no, the triangle: vertices X, Y, Z. Side XY: length 2? Wait, no, the diagram: XY is length 2, angle at Y is \( 75^\circ \), angle at Z is \( 50^\circ \), side XZ is y. Wait, let's re - label. Let's say: in triangle XYZ, angle at Y: \( 75^\circ \), angle at Z: \( 50^\circ \), side XY: length 2 (opposite angle Z), side XZ: length y (opposite angle Y). Then angle at X: \( 180 - 75 - 50 = 55^\circ \). Then Law of Sines: \( \frac{XY}{\sin(Z)}=\frac{XZ}{\sin(Y)} \). So \( \frac{2}{\sin(50^\circ)}=\frac{y}{\sin(75^\circ)} \). Then \( y=\frac{2\times\sin(75^\circ)}{\sin(50^\circ)} \). \( \sin(75^\circ)\approx0.9659 \), \( \sin(50^\circ)\approx0.7660 \). Then \( y=\frac{2\times0.9659}{0.7660}=\frac{1.9318}{0.7660}\approx2.5 \). Wait, that's one of the options. Oh, I see, I mixed up the opposite sides. So correct: side opposite angle Z (\( 50^\circ \)) is XY (length 2), side opposite angle Y (\( 75^\circ \)) is XZ (length y). So Law of Sines: \( \frac{y}{\sin(75^\circ)}=\frac{2}{\sin(50^\circ)} \). So \( y = \frac{2\times\sin(75^\circ)}{\sin(50^\circ)} \). Calculating: \( \sin(75^\circ)\approx0.9659 \), \( \sin(50^\circ)\approx0.7660 \). \( 2\times0.9659 = 1.9318 \). \( 1.9318\div0.7660\approx2.5 \).

Answer:

2.5 units