To solve this, we need to use the Law of Cosines. GIven a triangle ABC:
The Law of Cosines tells us:
[tex]a^2=b^2+c^2-2bc\cdot\cos(A)[/tex]
In the problem, we have 3 locations: Times Square (TS), Central Park (CP) and the Empire State Building (ESB). And we know the distances between them:
TS to CP = 2 miles
TS to ESB = 0.9 miles
CP to ESB = 1.3 miles
Let's make a diagram:
And using the Law of Cosines:
[tex]1.3=0.9^2+2^2-2\cdot0.9\cdot2\cdot\cos(\theta)[/tex]
And solve:
[tex]\begin{gathered} 1.3^2=0.81+4-3.6\cos(\theta) \\ . \\ 1.69=4.81-3.6\cos(\theta) \\ . \\ 3.6\cos(\theta)=4.81-1.69 \\ . \\ \cos(\theta)=\frac{3.12}{3.6}=\frac{13}{15} \\ . \\ \theta=\cos^{-1}(\frac{13}{15})\approx29.92643 \end{gathered}[/tex]
Thus, the correct answer is the second option, 29.926º
