Answer:
The bearing needed to navigate from island B to island C is approximately 38.213º.
Step-by-step explanation:
The geometrical diagram representing the statement is introduced below as attachment, and from Trigonometry we determine that bearing needed to navigate from island B to C by the Cosine Law:
[tex]AC^{2} = AB^{2}+BC^{2}-2\cdot AB\cdot BC\cdot \cos \theta[/tex] (1)
Where:
[tex]AC[/tex] - The distance from A to C, measured in miles.
[tex]AB[/tex] - The distance from A to B, measured in miles.
[tex]BC[/tex] - The distance from B to C, measured in miles.
[tex]\theta[/tex] - Bearing from island B to island C, measured in sexagesimal degrees.
Then, we clear the bearing angle within the equation:
[tex]AC^{2}-AB^{2}-BC^{2}=-2\cdot AB\cdot BC\cdot \cos \theta[/tex]
[tex]\cos \theta = \frac{BC^{2}+AB^{2}-AC^{2}}{2\cdot AB\cdot BC}[/tex]
[tex]\theta = \cos^{-1}\left(\frac{BC^{2}+AB^{2}-AC^{2}}{2\cdot AB\cdot BC} \right)[/tex] (2)
If we know that [tex]BC = 7\,mi[/tex], [tex]AB = 8\,mi[/tex], [tex]AC = 5\,mi[/tex], then the bearing from island B to island C:
[tex]\theta = \cos^{-1}\left[\frac{(7\mi)^{2}+(8\,mi)^{2}-(5\,mi)^{2}}{2\cdot (8\,mi)\cdot (7\,mi)} \right][/tex]
[tex]\theta \approx 38.213^{\circ}[/tex]
The bearing needed to navigate from island B to island C is approximately 38.213º.
