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### 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.

Hindus are the first to introduce sine and cosine instead of chords.

The Arabs worked with sine and cosine.

Hipparchus developed the first known table of chords.

Europeans translated theory into Latin.

Ptolemy is the next author of a book of chords.

Around 100 AD

140 BC

980 AD

100 AD

1500’s

500 AD

Hipparchus

- 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.
- 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

- 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 to 90° in quarter degree intervals.

equation used to calculate chord length

Appearance of sine and cosine

- The first appearance was in the work of the Hindus.
- half chords in a table = sin tables now
- 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.
- 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

- 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:
- Edmund Gunter: co-sinus
- Francois Viete: sinus residuae
- Cavalieri: Si 2
- William Oughtred: s co arc
- John Wallis: S

Tangent and cotangent

- 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

- Bonaventura Cavalieri: Ta and Ta 2
- William Oughtred: t arc and t co arc
- 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

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

- 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

- 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

- 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.

Applications to Physics

- 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:
- motion, distance, astronomy, and mechanics, vectors

“Demands of Trigonometry in Physics are enormous. So many problems in physics can be solved only by Trigonometry. Mostly all sections of Physics needs Trigonometry.” -Science.org

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 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

- "The Trigonometric Functions." Trigonometric Functions. N.p., n.d. Web. 02 Mar. 2014.
- "Hipparchus and Trigonometry." Starry Messenger: Hipparchus and Mathematical Techniques. N.p., n.d. Web. 02 Mar. 2014.
- The Editors of Encyclopædia Britannica. "Menelaus of Alexandria (Greek Mathematician)."Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Web. 02 Mar. 2014.
- "Ptolemy\'s Table of ChordsTrigonometry in the Second Century." Ptolemy\'s Table of Chords. N.p., n.d. Web. 02 Mar. 2014.
- "Trigonometry." Wikipedia. Wikimedia Foundation, 28 Feb. 2014. Web. 02 Mar. 2014. <http://en.wikipedia.org/wiki/Trigonometry>.
- "TRIGONOMETRY." Trigonometry. N.p., n.d. Web. 02 Mar. 2014. <https://www.physics.uoguelph.ca/tutorials/trig/trigonom.html>.

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