Question

Draw a diagram and solve using the Law of Cosines or the Law of Sines.

An airplane leaves an airport and flies due West 150 miles and then 150 miles in the direction S 39.67° W. How far is the plane from the airport? Round to the nearest mile.

Answer 1

Well, I figured this out on my own. Here's my process:

Because we don't have any matching angles, (when they're plotted on a triangle, you have a, b, and C) you have to use the Law of Cosines.

To find side c:
\[c^2 = a^2 + b^2 - 2ab \cos C\]Plug in the values
\[c^2 = 150^2 + 230^2 - 2(150)(230)\cos39.67^{o}\]And solve
\[22500+52900-96000 \cos 39.67^o\]\[c^2 =22288.3664\]\[c=149.2\]

Answer 2

To find Angle A:
\[a^2=b^2=c^2-2bc \cos A\]Substitute the values in:
\[150^2=230^2+149.2^2-2(230)(149.2) \cos A\]Move the cosine to the other side of the equation
\[\ A = cos^{-1} \left( \frac{ 230^2+149.2^2-150^2 }{ 2(230)(149.2) } \right)\]\[A=39.8\]

Answer 3

To find angle B:
\[b^2=a^2+c^2-2ac \cos B\]Move the cosine to the other side of the equation:
\[B=\cos^{-1} \left( \frac{ 150^2+149.2^2-230^2 }{ 2(150)(149.2) } \right)\]\[B=100.4\]