Practical_Antenna_Handbook_0071639586

A p p e n d i x A : U s e f u l M a t h 719
tanα = a b
or a = b tanβ (A.4.3e)
tanβ = b a
or b = a tanβ (A.4.3f)
Substituting in Eqs. (A.4.1), (A.4.3a), and (A.4.3.c), we can say:
h 2 = a 2 + b 2 = h 2 sin
2 α + h
2 cos
2 α
(A.4.4)
or
2 2
1 = sin α + cos α (A.4.5)
Suppose we draw a circle to represent the rotation of a chair on a Ferris wheel. Inside
that circle we draw a right triangle whose hypotenuse is equal to the radius of the
circle and is connected to the chair at one end, as shown in Fig. A.4.3. As the chair travels
the circumference of the circle, the right triangle we draw at each instant is a different
shape but we can always draw a right triangle. Because h, the hypotenuse, is equal
to the radius of the circle traversed by the chair, it is always constant. But sides a and b
are always changing. As an example: When the chair reaches the top of the Ferris wheel,
side b of the right triangle goes to 0 and side a is the same length as the hypotenuse h.
At that point, sin a = 0 and cos a = 1, agreeing with Eq. (A.4.5).
h'
a'
a''
b''
h''
b'
h
b
a
Figure A.4.3 Circular motion and the right triangle.