Answer:
The angle W is approximately 7°.
Step-by-step explanation:
Since angle X is adjacent to sides y and w and opposite to side x, we calculate the length of side x by Law of the Cosine:
[tex]x = \sqrt{y^2+w^2-2\cdot y\cdot w \cdot \cos X}[/tex] (1)
Where:
[tex]y, z[/tex] - Side lengths, in centimeters.
[tex]W[/tex] - Angle, in sexagesimal degrees.
If we know that [tex]y = 690\,cm[/tex], [tex]w = 440\,cm[/tex] and [tex]X = 163^{\circ}[/tex], then the length of the side x is:
[tex]x = \sqrt{y^2+w^2-2\cdot y\cdot w \cdot \cos X}[/tex]
[tex]x\approx 1118.199\,cm[/tex]
By Geometry we know that sum of internal angles in a triangle equals 180°. If X is an obtuse, then Y and W are both acute angles. By Law of the Sine we find angle W:
[tex]\frac{x}{\sin X}= \frac{w}{\sin W}[/tex] (2)
[tex]\sin W = \left(\frac{w}{x} \right)\cdot \sin X[/tex]
[tex]W = \sin^{-1}\left[\left(\frac{w}{x} \right)\cdot \sin X\right][/tex]
If we know that [tex]X = 163^{\circ}[/tex], [tex]w = 440\,cm[/tex] and [tex]x\approx 1118.199\,cm[/tex], then the angle W is:
[tex]W = \sin^{-1}\left[\left(\frac{w}{x} \right)\cdot \sin X\right][/tex]
[tex]W \approx 6.606^{\circ}[/tex]
Hence, the angle W is approximately 7°.
