103 Trigonometry Problems

1. Trigonometric Fundamentals 51
circle centered at C with radius CA; that is, ω denotes the boundary of the base of
the candle.
Let B be the foot of the perpendicular line segment from D to the circle ω, and
assume that ̸ OCB = θ. Because circle ω has radius 1, |ÔB|, the length of arc OB,is
θ. (This is why we use radian measure for θ.) Then B 1 = (θ, 0) and D 1 = (θ, y) with
y = BD. Let F be the foot of the perpendicular line segment from B to segment AC.
Then CF = cos θ, and AF = 1 + cos θ. Note that A, Q, C, F , and O are coplanar,
and ̸ OAQ = 45 ◦ . Point E is on segment AQ such that EF ⊥ AO. Consequently,
̸ AEF = ̸ OAQ = 45 ◦ and ̸ AFE = 90 ◦ , implying that the right triangle AEF
is isosceles, with AF = EF. It is not difficult to see that BDEF is a rectangle.
Hence BD = EF = AF = 1 + cos θ. We conclude that D 1 = (θ, 1 + cos θ); that
is, D 1 lies on the curve y = 1 + cos x.
Q1
2
1.5
D1
1
0.5
A1
-3 -2 -1 1 2 3
O1
-0.5
B1 A2
Figure 1.50.
Finally, had we not cut off the bottom of the candle, the equation of the curve
would have been y = 2 + cos x.
Three Dimensional Coordinate Systems
We view Earth as a sphere, with radius 3960 miles. We will set up two kinds of 3-D
coordinate systems to describe the positions of places on Earth.