Answer:
There are two choices for angle Y: [tex]Y \approx 54.987^{\circ}[/tex] for [tex]XZ \approx 15.193[/tex], [tex]Y \approx 27.008^{\circ}[/tex] for [tex]XZ \approx 8.424[/tex].
Step-by-step explanation:
There are mistakes in the statement, correct form is now described:
In triangle XYZ, measure of angle X = 49°, XY = 18 and YZ = 14. Find the measure of angle Y:
The line segment XY is opposite to angle Z and the line segment YZ is opposite to angle X. We can determine the length of the line segment XZ by the Law of Cosine:
[tex]YZ^{2} = XZ^{2} + XY^{2} -2\cdot XY\cdot XZ \cdot \cos X[/tex] (1)
If we know that [tex]X = 49^{\circ}[/tex], [tex]XY = 18[/tex] and [tex]YZ = 14[/tex], then we have the following second order polynomial:
[tex]14^{2} = XZ^{2} + 18^{2} - 2\cdot (18)\cdot XZ\cdot \cos 49^{\circ}[/tex]
[tex]XZ^{2}-23.618\cdot XZ +128 = 0[/tex] (2)
By the Quadratic Formula we have the following result:
[tex]XZ \approx 15.193\,\lor\,XZ \approx 8.424[/tex]
There are two possible triangles, we can determine the value of angle Y for each by the Law of Cosine again:
[tex]XZ^{2} = XY^{2} + YZ^{2} - 2\cdot XY \cdot YZ \cdot \cos Y[/tex]
[tex]\cos Y = \frac{XY^{2}+YZ^{2}-XZ^{2}}{2\cdot XY\cdot YZ}[/tex]
[tex]Y = \cos ^{-1}\left(\frac{XY^{2}+YZ^{2}-XZ^{2}}{2\cdot XY\cdot YZ} \right)[/tex]
1) [tex]XZ \approx 15.193[/tex]
[tex]Y = \cos^{-1}\left[\frac{18^{2}+14^{2}-15.193^{2}}{2\cdot (18)\cdot (14)} \right][/tex]
[tex]Y \approx 54.987^{\circ}[/tex]
2) [tex]XZ \approx 8.424[/tex]
[tex]Y = \cos^{-1}\left[\frac{18^{2}+14^{2}-8.424^{2}}{2\cdot (18)\cdot (14)} \right][/tex]
[tex]Y \approx 27.008^{\circ}[/tex]
There are two choices for angle Y: [tex]Y \approx 54.987^{\circ}[/tex] for [tex]XZ \approx 15.193[/tex], [tex]Y \approx 27.008^{\circ}[/tex] for [tex]XZ \approx 8.424[/tex].
