Trigonometric
Functions
Max Saeger and Kyle Ruescher
Beginnings
Menelaus developed more
tables of chords, but his work
didn’t last, he had a greater
influence on spherical
trigonometry.
Hipparchus
developed the
first known table
of chords.
140 BC
Ptolemy is the next
author of a book of
chords.
100 AD
Around
100 AD
Hindus are
the first to
introduce
sine and
cosine
instead of
chords.
500 AD
The
Arabs
worked
with sine
and
cosine.
980 AD
Europeans
translated
theory into
Latin.
1500’s
Hipparchus
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When Hipparchus was calculating the eccentricity of the orbits of the moon
and sun, he needed a method to do so. This method was chords.
o chord(A) = r(2 sin(A/2))
Chords were the first method of calculating trigonometric functions. 60
units of chords were used originally.
The challenge was to find the length of the chord with a given angle
between it. The formula used was 2sin(x/2), obviously not in terms of sin,
but chords.
There is little known about chords in the time of Hipparchus because the
books he recorded his findings in were destroyed.
Menelaus
•
In about 100 AD, Menelaus produced a table of chords as well. His book
on chords was also lost, but he is well known for his work in spherical
trigonometry.
● The form of this theorem for plane
triangles says that: if the three
sides of a triangle are crossed by
a straight line (one of the sides is
extended beyond its vertices),
then the product of three of the
nonadjacent line segments thus
formed is equal to the product of
three other line segments.
Claudius Ptolemy
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The next man to develop a table of chords.
He divided the circle into 360 degrees and the diameter into 120 different
parts. Ptolemy was able to calculate the corresponding chord length for
every central angle up to 180° in half-degree intervals.
● The table of chords as
compiled by Ptolemy is
equivalent to a table of
sines for every angle up
equation
used to
to 90° in quarter degree
calculate
intervals.
chord
length
Appearance of sine and cosine
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The first appearance was in the work of the Hindus.
o half chords in a table = sin tables now
o Jya was the word they used for sin
The Arabs worked with sine and cosine as well, calling sine jiba, which
was adopted from the Hindus, and did not have a meaning, it was just a
syllable.
o Soon after, jiba was changed to jaib which did have a meaning: fold.
When European authors translated the word jaib into Latin, it became
sinus, which also means fold.
The term “sine” was not originally accepted by authors, and the first time it
was used was in 1624 in a drawing by Edmund Gunter.
Rheticus and Copernicus
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All of Copernicus’s trigonometry relevant to astronomy
was published by Rheticus in 1542.
Rheticus also produced substantial amounts on sine
and cosines which was published after his death.
Abbreviations of cosine
•
The cosine took on many different forms, and had a similar course of
development as sine, leading up to complete acceptance by the authors
and mathematicians of the day. Here are some examples of the various
names for cosine:
o Edmund Gunter: co-sinus
o Francois Viete: sinus residuae
o Cavalieri: Si 2
o William Oughtred: s co arc
o John Wallis: S
Tangent and cotangent
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The tangent and cotangent developed slightly differently than the sine and
cosine.
Instead of helping to find angles, they were used to find the heights of an
object from the length of the shadow cast by the object.
The first known tables of shadows were produced by the Arabs around
860.
The name tangent was first used by Thomas Fincke in 1583 and the term
cotangents was first used by Edmund Gunter in 1620.
Abbreviations of tangent and cotangent
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Bonaventura Cavalieri: Ta and Ta 2
•
William Oughtred: t arc and t co arc
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John Wallis: T and t
Albert Girard: used tan, but wrote it on top of the angle
Secant and cosecant
•
Were not used by early
astronomers or surveyors,
but rather arrived when
navigators around the 15th
century started to prepare
tables.
graph of secant and
cosecant
Graph of sine and cosine
Graph of tangent and cotangent
Understanding Triangles
•
Right triangles are vital to trigonometry. A right triangle,
for example ( a triangle with a 90 degree angle) has
three sides. Two legs (labeled a and b) and a
hypotenuse (labeled c) directly across the triangle from
the right angle.
c(hypotenuse)
a(leg)
b(leg)
Sine, Cosine, Tangent
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What is meant when we say sine? Aren’t those the
things with the speed limits on the side of the road?
No, those are signs not sines. A sine, cosine, or tangent
is used to describe relationships between angles and
ratios of the lengths of their sides in a triangle.
With this information we can calculate various important
unknown angle or side lengths.
Sine, Cosine, Tangent
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The sine of an angle is defined as the ratio of the side opposite of the
angle in use to the hypotenuse. The Cosine of an angle is slightly different.
Cosine of an angle is defined as the ratio of adjacent to hypotenuse.
Tangent of an angle can be defined as either opposite to adjacent or sine
to cosine.
Mathematically written: TRIG(angle)=ratio
sin(angle)=Opposite/Hypotenuse
cos(angle)=Adjacent/Hypotenuse
tan(angle)=opposite/adjacent=sin(angle)/cos(angle)
Other important functions coming from
Sine, Cosine, Tangent
Cosecant(angle)=1/sin(angle)
Secant(angle)=1/cos(angle)
Cotangent(angle)=1/tan(angle)=cos(angle)/sin
(angle)
Unit Circle
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The unit circle may be as important to trig as
differentiating is to calculus. The unit circle is a circle
constructed with a radius of 1. This radius of 1 is the
hypotenuse of any angle in the circle. Furthermore we
can use pythagorean Theorem to calculate ratios that
specific angles give out. This is helpful because it
becomes a standard way to use trig to solve problems
without having to make a triangle every time.
Unit Circle Of Major Angles
Applications to Physics
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Okay great, I’m probably starting to sound like your math teacher. Why does it even matter?
There are so many different applications in life, math, and EVERY science. Some very
interesting particular applications to physics include but are not nearly limited to:
Optics
Statics
vector problem solving
rotational motion
gravitation
motion in a plane
wave mechanics
Even Problems in YOUR life, that’s right. Dr. Hopkins has used many applications of trig in class
to describe:
o
motion, distance, astronomy, and mechanics, vectors
“Demands of Trigonometry in Physics are enormous. So many problems in physics can be solved only
Impact of Trig on Culture
There is trig all around you, we do not even
notice it, it has become such a commonplace
in our lives and culture, especially as a first
world country. Without trig:
• The building you’re in would most likely not be standing.
• Electronic devices would need a new way to operate without the ability to
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use open and closed systems.
The roads and bridges would collapse, or be extremely unorganized
We probably would never get a man to the moon.
We wouldn’t know nearly as much about the planets and stars.
The
Canadarm 2 robotic manipulator on the
international space station is operated by
controlling the angles of its joints. Calculating
the final position of the astronaut at the end
of the arm requires repeated use of
trigonometric functions of those angles.
Works cited
1. "The Trigonometric Functions." Trigonometric Functions. N.p., n.d. Web. 02
Mar. 2014.
2. "Hipparchus and Trigonometry." Starry Messenger: Hipparchus and
Mathematical Techniques. N.p., n.d. Web. 02 Mar. 2014.
3. The Editors of Encyclopædia Britannica. "Menelaus of Alexandria (Greek
Mathematician)."Encyclopedia Britannica Online. Encyclopedia Britannica, n.d.
Web. 02 Mar. 2014.
4. "Ptolemy's Table of ChordsTrigonometry in the Second Century." Ptolemy's Table
of Chords. N.p., n.d. Web. 02 Mar. 2014.
5. "Trigonometry." Wikipedia. Wikimedia Foundation, 28 Feb. 2014. Web. 02 Mar.
2014. <http://en.wikipedia.org/wiki/Trigonometry>.
6. "TRIGONOMETRY." Trigonometry. N.p., n.d. Web. 02 Mar. 2014.