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The measure of the side [tex]x[/tex] is approximately 33 and the measure of angle [tex]Y[/tex] is approximately 12.321°.
In this problem we must apply the law of cosine and the law of sine to determine the angle Y:
[tex]x = \sqrt{y^{2}+z^{2}-2\cdot y\cdot z\cdot \cos X}[/tex]
[tex]x = \sqrt{15^{2}+19^{2}-2\cdot (15)\cdot (19)\cdot \cos 152^{\circ}}[/tex]
[tex]x \approx 33[/tex]
[tex]\frac{\sin Y}{y} = \frac{\sin X}{x}[/tex]
[tex]Y = \sin^{-1}\left(\frac{y}{x}\cdot \sin X \right)[/tex]
[tex]Y = \sin^{-1}\left(\frac{15}{33}\cdot \sin 152^{\circ} \right)[/tex]
[tex]Y \approx 12.321^{\circ}[/tex]
The measure of the side [tex]x[/tex] is approximately 33 and the measure of angle [tex]Y[/tex] is approximately 12.321°. [tex]\blacksquare[/tex]
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