103 Trigonometry Problems

1. Trigonometric Fundamentals 57
Example 1.21. Monica and Linda traveled from Shanghai, China (east 121 ◦ , north
31 ◦ ), to Albany, NewYork (west 73 ◦ , north 42 ◦ ), to visit their friend Hilary. Estimate
the distance of Monica and Linda’s trip.
Solution: In general, let A = (3960,α 1 ,β 1 ) and B = (3960,α 2 ,β 2 ). Then
A = 3960(cos α 1 cos β 1 , sin α 1 cos β 1 , sin β 1 )
and
B = 3960(cos α 2 cos β 2 , sin α 2 cos β 2 , sin β 2 ).
Let θ = ̸ AOB. Then by the vector form of the law of cosines, wehave
−→
OA · −→ OB
cos θ =
| −→ OA|| −→ OB|
= cos α 1 cos β 1 cos α 2 cos β 2 + sin α 1 cos β 1 sin α 2 cos β 2 + sin β 1 sin β 2 .
For this problem, we have α 1 = 121 ◦ , β 1 = 31 ◦ , α 2 =−73 ◦ , and β 2 = 42 ◦ .
Plugging these values into the above equation gives cos θ ≈ −0.273, implying
that θ ≈ 105.870 ◦ . Hence the distance between Shanghai and Albany is about
θ
360 ◦ · 2π · 3960 ≈ 7317.786 miles along Earth’s surface.
Where Are You?
During a long flight, the screens in the plane cabin often show the position of the
plane and the trajectory of the trip. How is it done? Or, how does a GPS (Global
Positioning System) work? Let’s give a baby tour of this subject.
Example 1.22. Monica and Linda traveled from Shanghai, China (east 121 ◦ , north
31 ◦ ), toAlbany, NewYork (west 73 ◦ , north 42 ◦ ), to visit their friend Hilary. Monica’s
hometown, Billrock, is four-fifths of the way along their trip. Find the spherical
coordinates of the town of Billrock.
C
C
A
A
D
B
B
k
(1- k)
E
O
O
Figure 1.57.