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The direction of Beatriz relative to the starting point of her trip is approximately [tex]278.806^{\circ}[/tex].
After a careful reading of the statement, we find that final position ([tex]\vec r[/tex]) by the end of the second day is found by means of this vector sum:
[tex]\vec r = \vec r_{1} + \vec r_{2}[/tex] (1)
Where:
If we know that [tex]\vec r_{1} = (200\,km \cdot \cos 300^{\circ}, 200\,km\cdot \sin 300^{\circ})[/tex] and [tex]\vec r_{2} = (150\,km\cdot \cos 250^{\circ}, 150\,km\cdot \sin 250^{\circ})[/tex], then final position of Beatriz relative to origin is:
[tex]\vec r = (200\,km\cdot \cos 300^{\circ}, 200\,km\cdot \sin 300^{\circ})+(150\,km \cdot \cos 250^{\circ}, 150\,km\cdot \sin 250^{\circ})[/tex]
[tex]\vec r = (48.670, -314.159)\,[km][/tex]
And the direction relative to the starting point ([tex]\theta[/tex]), in degrees, is found by following inverse trigonometric relation:
[tex]\theta = \tan^{-1} \frac{r_{y}}{r_{x}}[/tex] (2)
If we know that [tex]r_{x} = 48.670\,km[/tex] and [tex]r_{y} = -314.159\,km[/tex], then the direction of Beatriz relative to the starting point of her trip is:
[tex]\theta = \tan^{-1} \left(\frac{-314.159\,km}{48.670\,km} \right)[/tex]
[tex]\theta \approx 278.806^{\circ}[/tex]
The direction of Beatriz relative to the starting point of her trip is approximately [tex]278.806^{\circ}[/tex]. [tex]\blacksquare[/tex]
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