An Example: Height of A Sailboat Mast

Trigonometry is a branch of mathematics that studies relationships between the sides and angles
of triangles. Trigonometry is found all throughout geometry, as every straight-sided shape may be
broken into as a collection of triangles. Further still, trigonometry has astoundingly intricate
relationships to other branches of mathematics, in particular complex numbers, infinite series,
logarithms and calculus.
The word trigonometry is a 16th-century Latin derivative from the Greek words for triangle (trigōnon)
and measure (metron). Though the field emerged in Greece during the third century B.C., some of
the most important contributions (such as the sine function) came from India in the fifth century
A.D. Because early trigonometric works of Ancient Greece have been lost, it is not known whether
Indian scholars developed trigonometry independently or after Greek influence. According to Victor
Katz in “A History of Mathematics (3rd Edition)” (Pearson, 2008), trigonometry developed primarily
from the needs of Greek and Indian astronomers.
An example: Height of a sailboat mast
Suppose you need to know the height of a sailboat mast, but are unable to climb it to measure. If
the mast is perpendicular to the deck and top of the mast is rigged to the deck, then the mast, deck
and rigging rope form a right triangle. If we know how far the rope is rigged from the mast, and the
slant at which the rope meets the deck, then all we need to determine the mast’s height is
trigonometry.
For this demonstration, we need to examine a couple ways of describing “slant.” First is slope,
which is a ratio that compares how many units a line increases vertically (its rise) compared to how
many units it increases horizontally (its run). Slope is therefore calculated as rise divided by run.
Suppose we measure the rigging point as 30 feet (9.1 meters) from the base of the mast (the run).
By multiplying the run by the slope, we would get the rise — the mast height. Unfortunately, we
don’t know the slope. We can, however, find the angle of the rigging rope, and use it to find the
slope. An angle is some portion of a full circle, which is defined as having 360 degrees. This is
easily measured with a protractor. Let’s suppose the angle between the rigging rope and the deck
is 71/360 of a circle, or 71 degrees.
We want the slope, but all we have is the angle. What we need is a relationship that relates the
two. This relationship is known as the “tangent function,” written as tan(x). The tangent of an angle
gives its slope. For our demo, the equation is: tan(71°) = 2.90. (We'll explain how we got that
answer later.)
This means the slope of our rigging rope is 2.90. Since the rigging point is 30 feet from the base of
the mast, the mast must be 2.90 × 30 feet, or 87 feet tall. (It works the same in the metric system:
2.90 x 9.1 meters = 26.4 meters.)
Sine, cosine and tangent

Depending on what is known about various side lengths and angles of a right triangle, there are two
other trigonometric functions that may be more useful: the “sine function” written as sin(x), and the
“cosine function” written as cos(x). Before we explain those functions, some additional terminology
is needed. Sides and angles that touch are described as adjacent. Every side has two adjacent
angles. Sides and angles that don’t touch are described as opposite. For a right triangle, the side
opposite to the right angle is called the hypotenuse (from Greek for “stretching under”). The two
remaining sides are called legs.
Usually we are interested (as in the example above) in an angle other than the right angle. What
we called “rise” in the above example is taken as length of the opposite leg to the angle of interest;
likewise, the “run” is taken as the length of the adjacent leg. When applied to an angle measure,
the three trigonometric functions produce the various combinations of ratios of side lengths.
In other words:
 The tangent of angle A = the length of the opposite side divided by the length of the adjacent side
 The sine of angle A = the length of the opposite side divided by the length of the hypotenuse
 The cosine of angle A = the length of the adjacent side divided by the length of the hypotenuse
From our ship-mast example before, the relationship between an angle and its tangent can be
determined from its graph, shown below. The graphs of sine and cosine are included as well.

An Example: Height of A Sailboat Mast

Uploaded by

Copyright:

Available Formats

An Example: Height of A Sailboat Mast

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

An Example: Height of A Sailboat Mast

Uploaded by

Copyright:

Available Formats

Trigonometry is a branch of mathematics that studies relationships between the sides and angles

Sine, cosine and tangent

You might also like