Trigonometry in Cable Television
(adapted from “Tech Book,” originally published in Communications Technology, February 1987, by Ron Hranac and Bruce Catter)
In any right, or 90º, triangle (Figure 1), if angle (θ) is between 0º and 90º, the sine, cosine and tangent are
ratios of the sides of that triangle, as described by the most common trigonometric ratios:
the sine of angle (θ) = side b/side c
the cosine of angle (θ) = side a/side c
the tangent of angle (θ) = side b/side a
c
b
90°
Angle
Angle
a
Figure 1. Right Triangle
From these ratios, formulas can be derived (Figure 2) to determine solutions to problems that occasionally
crop up in CATV engineering. Examples of the application of some of these formulas follow.
Figure 2. Trigonometric formulas
Problem 1:
A programmer’s move to another satellite requires the installation of a new dish at your headend, but the
configuration of the site only will allow installation behind the existing dish (Figure 3). Assuming the existing
dish is 18 feet high, the minimum antenna pointing elevation to any satellite in the visible geosynchronous
arc from your location is 32°, and the lower rim of the new antenna will be 3 feet off the ground at that
elevation, how close behind the existing dish can the new one be installed?
Figure 3. New Satellite Dish Installation (Problem 1)
Solution:
Subtract the new dish’s lower rim height (3 feet at the minimum usable elevation) from the height of the
existing antenna. The result, 15 feet, is the height of the “b” side of the triangle being used to solve this
problem. Then, use the formula
a = b / tan θ
= 15 feet / tan 32°
= 15 / 0.6249
= 24 feet
Problem 2:
Your company has just purchased a cable system, and you have been asked to verify the height of the tower
at the new system’s headend. With an accurate measuring wheel, you locate a spot 200 feet from the tower
(Figure 4). From that spot, using an Abney level or a surveying transit, you measure the angle from the base
to the top of the tower, and find it to be 27°. What is the height of the tower?
Figure 4. Verify Tower Height (Problem 2)
Solution:
Use the formula
b = a (tan θ)
=200 feet (tan 27°)
= 200 (0.5095)
= 101.9 feet
Note: The measurement of the angle θ must be made at the same level
of the base of the structure being checked, or the calculated height will
be inaccurate. If this is not possible, then additional calculations, such
as those in Problem 3, will be necessary.
Figure 5. Calculate Tower Height for Unlevel Ground (Problem 3)
Problem 3:
To calculate the height of the tower in this situation, first measure c2 with an accurate measuring wheel; also
measure angle θ2 from level down to the base of the tower, and angle θ1 from level up to the top of the tower.
(Assume c2 = 200 feet, θ2 = 8°, and θ1 = 25° for this example.)
(Problem 3 continued on next page)
Calculate b2 with the formula
b2 = c2 (sin θ2)
= 200 feet (sin 8°)
= 200 (0.1392)
b2 = 27.8 feet
Calculate a with the formula
a = b2 / tan θ2
= 27.8 feet / tan 8°
= 27.8 / 0.1405
a = 197.8 feet
b1 is found using the formula
b1 = a(tan θ1)
= 197.8 feet (tan 25°)
= 197.8 (0.4663)
b1 = 92.2 feet
The height of the tower, b1 + b2, is 120 feet.
Note: There are commercially available devices such as a topographic Abney level that can be used for
measuring elevation angles, and they are relatively inexpensive. An inclinometer (or clinometer) can be
home-made from a simple protractor and a weighted string attached to the protractor (like we did in middle
school). YouTube has many videos on how to make and use the home-made inclinometer and if used
correctly it will be accurate enough for most applications. Have fun!
Trig Basic Functions “SohCahToa”
The mnemonic “SohCahToa” can help you remember the trigonometric formulae.
Soh = Sine θ =
opposite (length of b)
hypotenuse (length of c)
Cah = Cosine θ = adjacent (length of a)
hypotenuse (length of c)
Toa = Tangent θ = opposite (length of b)
adjacent (length of a)
QUIZ IN PROGRESS
To assess your understanding of Trigonometry in Cable Television please take this short quiz.
Problem 4:
Now it’s your turn to practice a few calculations. Your company has just purchased yet another cable
system, and once again you must verify the height of the tower. With an accurate measuring wheel, you
locate a spot 150 feet from the tower. What is the height of the tower?
(Assume c2 = 150 feet, θ2 = 10°, and θ1 = 30° for this example.)
Calculate b2 with the formula
b2 = c2 (sin θ2)
= 150 feet (sin 10°)
= 150 x _________
b2 = _______ feet
Calculate a with the formula
a = b2 / tan θ2
= _______ feet / tan 10°
= _______ / _______
a = _______ feet
b1 is found using the formula
b1 = a(tan θ1)
= _______ feet (tan 30°)
= _______ x _______
b1 = _______ feet
The height of the tower, b1 + b2 = _______ + _______ = _______ feet.
(Problem 4 continued on next page)
Solution to Problem 4
Calculate b2 with the formula
b2 = c2 (sin θ2)
= 150 feet (sin 10°)
= 150(0.17365)
b2 = 26.05 feet
Calculate a with the formula
a = b2 / tan θ2
= 26.1 feet / tan 10°
= 26.1 / 0.17633
a = 148.0 feet
b1 is found using the formula
b1 = a(tan θ1)
= 148.0 feet (tan 30°)
= 148.0 (0.57735)
b1 = 85.5 feet
The height of the tower, b1 + b2 = 85.5 + 26.05 = 111.5 feet.