User:Egm6322.s12.TEAM1.WILKS/Term Paper

Division
Since the beginning of time, humans have shared or divided their possessions, food and shelter in order to maintain balance and harmony in their lives. The concept of division of untold bounties is therefore a topic worth repeating due to its importance in our lives. In mathematic terms, division (÷) is a simple arithmetic operation. For instance, 8 ÷ 2 = 4 In other words, if I have 8lbs of fish to divide between two families, each will receive 4 lbs of fish. In formula form, if b is not zero, then a divided by b equals c, can be written as: a ÷ b = c Which can also be expressed as $$ \frac{a}{b}=c \ $$ In the above expression, a is called the dividend, b the divisor and c the quotient. Division leads to the concept of fractions being introduced. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. Rational numbers is any number that is an integer or it's remainder either terminates or repeats a sequence of the remainder. A term that satisfies this definition of a rational number may also be called commensurable. If a number does not follow the definition of a rational number, it is defined as irrational. In mathematics, an irrational number is any real number that cannot be expressed as a ratio $$ \frac{a}{b} \ $$, where a and b are integers, with b non-zero, with a terminating and/or a repeating remainder. Irrational numbers are known to incommensurable, i.e. meaning they share no measure in common. Perhaps the best-known irrational numbers are: the ratio of a circle's circumference to its diameter $$ \pi\ \ $$,  Euler's number $$ e \ $$, and the square root of two $$ \sqrt{2} \ $$.

Fractions
Fractions are mathematical formulae used to show a part of a whole number. As shown above, fractions are given by: $$ \frac{a}{b}=c \ $$ where a is called the dividend (i.e. part of the whole number), b the divisor (i.e. the whole number) and c the quotient (i.e. how many times the divisor can be multiplied into the whole number). Fractions are frequently used in engineering due to their usefullness in providing quick and standard units of measurement. English units of length are usually given in 32nds, such as: $$ \frac{8}{32}in = 0.25in \ $$ whereas the International System of Units (SI), also known as the metric system, is in a form of meters and since this sytem has a base untis of ten, these are usually given as tenths, such as: $$ \frac{5}{10}m = 0.5m \ $$.

Continued Fractions
Continued fractions are a special form of fractions that offer a useful means of expressing numbers and functions. Continued fractions are obtained through an iterative process of adding an integer plus a fraction, in which the divisor of the fraction is an integer plus a fraction. This iteration of the divisor may continue infinitely or may be a finite iteration depending on whether the number being represented is rational or irrational. Irrational numbers will have an infinite continued fraction representation, whereas rational numbers will have some finite or limited continued fraction. If the continued fraction expansion terminates after a number k partial quotients, the value of the term is notated as the "kth convergent" of the continued fraction. A periodic continued fraction is a form of the continued fraction expansion that leads to the terms of $$ a_i \ $$ and $$ b_i \ $$ repeatly cyclically. A number is defined as rational if and only if it can expressed as a simple finite continued fraction. A continued fraction may be expressed as: $$ r = a_0+\cfrac{b_1} {a_1+\cfrac{b_2} {a_2+\cfrac{b_3} { a_3+ \ddots } } } \ $$ Where $$ a_i \ $$ and $$ b_i \ $$ may be rational, real or complex numbers. A continued fraction is considered simple if all $$ b_i=1 \ $$. A continued fraction that contains a finite number of terms is known as a finite continued fraction. The terms of $$ a_i \ $$ in the above sample are known as partial quotients. An example of a simple finite continued fraction expansion of a rational number may be given as: $$ \frac{9}{16} = \cfrac{1} {1+\cfrac{1} {1+\cfrac{1} { 3+ \cfrac{1}{2} } } } \ $$ The compact notation for this continued fraction may be expressed as follows: $$ \frac{1}{1+}\frac{1}{1+}\frac{1}{3+}\frac{1}{2} \ $$ Note that the denominators of this compact notation follow the left hand side of the continued fraction expansion. The continued fraction for $$ \tfrac{9}{16} \ $$ as shown above was found using an iterative method. An easy way to view the iterative process that results in the continued fraction expansion is detailed in the following table:
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! colspan="7" |Find the continued fraction for .5625 (= $$ \tfrac{9}{16}$$) ! colspan="7" | Continued fraction form for $$ \tfrac{9}{16} = 0.5625 \ $$ is [0; 1, 1, 3, 2] As shown in the bottom row of the table above, another compact notation for the continued fraction may be represented as: $$x = [a_0; a_1, a_2, a_3] \;$$ where $$ a_1, a_2 \cdots \ $$ are the quotients of the continued fraction. If $$ a_0 \ $$ is an integer, it is normally separated from the quotients by a semicolon in this notation. Looking back at the table above, it may be noted that the quotients are given in the integer column of the table. The standard expression for a infinite continued fraction is given here: $$ a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{a_4+\cfrac{1}{\ddots}}}}} \ $$ where $$ a_0 \ $$ is an integer and $$ \left \{a_1, a_2, a_3, a_4 \cdots a_i\right \} \ $$ for $$ \left \{i=1,2, \cdots, n\right \} \ $$ as $$ \left \{ n \rightarrow \infty \ \right \} $$ are positive integers. Every irrational number may be expressed mathematically as an infinite continued fraction. Infinite continued fractions are important due to the approximating power of the initial segments. An infinite continued fraction converges if the sequence of convergents approaches a limit. If the sequence of convergents does not approach a limit, the continued fraction is divergent. A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction. For example, given the fraction $$ \frac{9}{16} \ $$ as calculated above, the first convergent would be the truncation of the continued fraction after one iteration: $$ \frac{9}{16} = \cfrac{1} {1+\cfrac{1}{1}} = 0.5 \ $$ As one can see, this answer is not as accurate as if one had taken two iterations, resulting in the following: $$ \frac{9}{16} = \cfrac{1} {1+\cfrac{1} {1+\cfrac{1} { 3 } } } \approx .571 \ $$ Finally this continued fraction converges as it reaches its terminal decimal point with the following: $$ \frac{9}{16} = \cfrac{1} {1+\cfrac{1} {1+\cfrac{1} { 3+ \cfrac{1}{2} } } } = .5625 \ $$ This nth convergent is also known as the nth approximant of a continued fraction. Each convergent of that continued fraction is in a sense the best possible rational approximation to that real number, for a given number of digits. Such a convergent is usually about as accurate as a finite decimal expansion having as many digits as the total number of digits in the nth numerator and nth denominator. The example of truncating a finite continued fraction as shown above shows how convergence works. The power of convergence becomes much greater whn dealing with an infinite continued fraction. Consider any continued fraction as given by: $$ \left [ a_0;a_1, a_2, \ldots \right ] \ $$ This continued fraction has an ith convergent defined by the continued fraction: $$ \left [ a_0,a_1, a_2, \ldots, a_i \right ] \ $$ The ith convergent for continued fractions is given by $$ \frac{p_i}{q_i} \ $$, where $$ p_i \ $$ and $$ q_i \ $$ can be calculated recursively by the following equations: $$ p_n=a_np_{n-1}+p_{n-2} \ $$ $$ q_n=a_nq_{n-1}+q_{n-2} \ $$ In the equations for $$ p_n \ $$ and $$ q_n \ $$ above, the initial conditions are: $$ p_0=a_0 \ $$, $$ q_0=1 \ $$, $$ p_1=a_0a_1+1 \ $$ and $$ q_1=a_1 \ $$
 * Step
 * Real Number
 * Integer part
 * Fractional part
 * Simplified
 * Reciprocal of $$f$$
 * Simplified
 * $$1$$
 * $$r = \frac{9}{16}\,$$
 * $$i = 0\,$$
 * $$f = 0 \tfrac{9}{16} - 0\,$$
 * $$= \tfrac{9}{16}\,$$
 * $$1/f = \tfrac{16}{9}\,$$
 * $$= 1 \tfrac{7}{9}\,$$
 * $$2$$
 * $$r =1 \tfrac{7}{9}\,$$
 * $$i = 1\,$$
 * $$f = 1 \tfrac{7}{9}- 1\,$$
 * $$= \tfrac{7}{9}\,$$
 * $$1/f = \tfrac{9}{7}\,$$
 * $$= 1 \tfrac{2}{7}\,$$
 * $$3$$
 * $$r = 1 \tfrac{2}{7}\,$$
 * $$i = 1\,$$
 * $$f = 1 \tfrac{2}{7}- 1\,$$
 * $$= \tfrac{2}{7}\,$$
 * $$1/f = \tfrac{7}{2}\,$$
 * $$= 3 \tfrac{1}{2}\,$$
 * $$4$$
 * $$r = 3 \tfrac{1}{2}\,$$
 * $$i = 3\,$$
 * $$f = 3 \tfrac{1}{2}- 3\,$$
 * $$= \tfrac{1}{2}\,$$
 * $$1/f = \tfrac{2}{1}\,$$
 * $$= 2\,$$
 * $$5$$
 * $$r = 2\,$$
 * $$i = 2\,$$
 * $$f = 2 - 2\,$$
 * $$= 0\,$$
 * STOP
 * $$5$$
 * $$r = 2\,$$
 * $$i = 2\,$$
 * $$f = 2 - 2\,$$
 * $$= 0\,$$
 * STOP
 * }

History of Continued Fractions
There are indications that continued fractions may have been used as early as 306 B.C.E.. As early as the fifth century A.D., a Hindu mathematician used a continued fraction to solve a linear equation. Most early continued fraction use was limited to specific examples and not generalized and widely used for more than a thousand years. Several early mathematicians, notably Rafael Bombelli and Pietro Cataldi provided good examples of continued fractions, but stopped short of fully investigating the topic. Continued fractions blossomed into the generalized continued fraction theory through the work of John Wallis (1616-1703). In his book Arithemetica Infinitorium, Wallis developed the identity: $$ \frac{ \pi\ }{4} = \frac{3*3*5*5*7*7*9 \cdots}{2*4*4*6*6*8*9\cdots} \ $$ Which the first president of the Royal Society, Lord Brouncker (1620-1684) transformed into: $$ \frac{ \pi\ }{4} = 1+ \cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} { 2+ \cfrac{2}{ \ddots } } } } \ $$ Lord Brouncker did not further develop the theory of continued fractions after this. John Wallis took the first major steps in recording his theory on the subject. In his book Opera Mathematica, Wallis first coined the term "continued fraction". In other mathematicians earlier works, continued fractions were coined as "anthyphairetic ratios". The Dutch mathematician and astronomer Christiaan Huygens was the first to develop a practical application of continued fractions. He wrote a paper explaining how to use the convergents of a continued fraction to find the best approximations for gear ratios. Much of the modern theory of continued fraction was developed by three major contributors to the field of mathematics: Leonard Euler, Johan Lambert and Joseph Louis Lagrange. In Euler's work titled De Fractionbous Continuis, it was shown that every rational number could be expressed as a simple finite continued fraction. Euler's contributions to the continued function of $$ e \ $$ are noted in the section on $$ e \ $$ in this paper. Johan Lambert worked alongside Euler and was able to generalize Euler's work on $$ e \ $$ to prove that both $$ e^x \ $$ and $$ \tan x \ $$ are irrational if $$ x \ $$ is rational. Other known contributors to the field of continued fractions include: Karl Jacobi, Charles Hermite, Karl Friedrich Gauss and Augustin Cauchy, which are common names in the field of mathematics.

Continued Fraction Expansion of pi
Pi also known by the symbol $$ \pi \ $$ is known as the constant ratio of the circumference of a circle to its diameter also given by the fraction: $$ \frac{cirumference}{diameter} \ $$ This ratio's significance and importance in history is difficult to put into words. The ratio for pi has existed as far back as mankind has records. The first recorded approximations for Pi were found in Babylon and Egypt. The ancient Babylonians inscribed their interpretation of Pi=3 on a tablet (circa 1900-1680 BCE) shown here: Over many centuries, more accurate values have been found for $$ \pi \ $$. "One early value was the Greek approximation 3 1-7, found by considering the circle as the limit of a series of regular polygons with an increasing number of sides inscribed in the circle. About the mid-19th cent. its value was figured to 707 decimal places and by the mid-20th cent. an electronic computer had calculated it to 100,000 digits. It would have taken a person working without error eight hours a day on a desk calculator 30,000 years to make this calculation; it took the computer eight hours. Although it has now been calculated to more than 200,000,000,000 digits, the exact value of π cannot be computed" For many common engineering uses, i.e. machining hardware, Pi, approximated to three or four decimal places suffices for many day to day calculations. For more exact usage, "a value of π to 40 digits would be more than enough to compute the circumference of a circle as large as the Milky Way galaxy to an error less than the size of a proton." Because Pi is an irrational number, there is no "exact" fraction that can represent the ratio, thus presented below are the three most commonly known generalized continued fraction expansions for the term Pi: $$ \pi = \cfrac{4}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}} = 3+\cfrac{1^2}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cfrac{9^2}{6+\ddots}}}}} = \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ddots}}}}} \approx 3.1416 $$ The first two continued fractions for Pi shown here are generally considered to be slow in terms of convergence. The first term requires $$ 3*10_n \ $$ terms to reach n-decimal precision. This means that for four place decimal precision, Pi=3.1416, the first term would require 120 terms of expansio to reach this precision. The second representation shown is still a very slow convergence, requiring nearly 50 terms for five decimals and nearly 120 for six. The third continued fraction expansion converges almost linearly for Pi. Convergence for this term is at least three terms or decimal places of precision per every four fractional terms calculated. The theory of continued fractions provides the sequence of the best rational approximations for the number $$ \pi \ $$ as given here: $$ \pi = 3+\cfrac{1}{6+\cfrac{9}{6+\cfrac{25}{6+\cfrac{49}{6+\cfrac{81}{6+\cfrac{121}{6+\ddots}}}}}}  $$ Since $$ \pi \ $$ is both irrational and transcendental, there is no integer the author can begin with to perform a calculation as shown in the continued fraction section. Given the limited tools at my disposal to measure a circle's diameter and circumference, the author is unable to accurately take measurements and approximate $$ \pi \ $$ using physical evidence. Using the method of trigonometry, the author calculates $$ \pi \ $$, by first calculating a very tiny piece of the circumference as the opposite side of a circle with radius =1; diameter =2. Length of small piece of circumference = $$ \sin {1*10^{-900}} \approx 1.745 * 10^{-902} \ $$ Given 360 degrees in a circle an having a very small length of the circumference, one calculates the number of small pieces needed to form the circumference. Number of small pieces needed to calculate perimeter = $$ \frac{360}{10^{-900}} \approx 3.6*10^902 \ $$ Circumference then is given by # of pieces required multiplied by Length of pieces = $$ 3.6*10^902 * 1.745 * 10^{-902} \approx 6.283185 \ $$ Given that $$ \pi \ = \frac{cirumference}{diameter} \ $$ then one can find $$ \pi \ $$ with $$ \frac{6.283185}{2} \ $$, where 2 is the diameter of the circle from which the cirumference was calculated. The author was able to calculate (on a Texas Instruments TI-92) $$ \pi = 3.141592654 \ $$, which is accurate to 8 decimal places. Using this solution, the author was able to "back door" a continued fraction solution, which is laid out in the table below:


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! colspan="7" |Find the continued fraction for $$ \pi $$) This table produces the continued fraction as defined below: $$ \pi\approx 3+ {1\over \displaystyle 7 + {1\over \displaystyle 15 + {1\over \displaystyle 1 + {1\over \displaystyle 293 + {1\over \displaystyle 12.05}}}}} \approx 3.141592654 $$ Which shows the continued fraction's accuracy to 9 decimal places.
 * Step
 * Real Number
 * Integer part
 * Fractional part
 * Simplified
 * Reciprocal of $$f$$
 * Simplified
 * $$1$$
 * $$r = \frac{31415927}{10000000}\,$$
 * $$i = 3\,$$
 * $$f = 3 \tfrac{1415927}{10000000} - 3\,$$
 * $$= \tfrac{1415927}{10000000}\,$$
 * $$1/f = \tfrac{10000000}{1415927}\,$$
 * $$= 7 \tfrac{625133}{1000000000}\,$$
 * $$2$$
 * $$r =7 \tfrac{625133}{1000000000}\,$$
 * $$i = 7\,$$
 * $$f = 7 \tfrac{625133}{1000000000}- 7\,$$
 * $$= \tfrac{625133}{1000000000}\,$$
 * $$1/f = \tfrac{1000000000}{625133}\,$$
 * $$= 15 \tfrac{9966}{10000}\,$$
 * $$3$$
 * $$r = 15 \tfrac{9966}{10000}\,$$
 * $$i = 15\,$$
 * $$f = 15 \tfrac{9966}{10000}- 15\,$$
 * $$= \tfrac{9966}{10000}\,$$
 * $$1/f = \tfrac{100000}{9966}\,$$
 * $$= 1 \tfrac{3412}{1000000}\,$$
 * $$4$$
 * $$r = 1 \tfrac{3412}{1000000}\,$$
 * $$i = 1\,$$
 * $$f = 1 \tfrac{3412}{1000000}- 1\,$$
 * $$= \tfrac{3412}{1000000}\,$$
 * $$1/f = \tfrac{1000000}{3412}\,$$
 * $$= 293 \tfrac{83}{1000}\,$$
 * $$5$$
 * $$r = 293 \tfrac{83}{1000} \,$$
 * $$i = 293\,$$
 * $$f = 293 \tfrac{83}{1000} - 293\,$$
 * $$= \tfrac{83}{1000}\,$$
 * $$1/f = \tfrac{1000}{83}\,$$
 * $$= 12.05\,$$
 * STOP
 * }
 * $$f = 293 \tfrac{83}{1000} - 293\,$$
 * $$= \tfrac{83}{1000}\,$$
 * $$1/f = \tfrac{1000}{83}\,$$
 * $$= 12.05\,$$
 * STOP
 * }
 * }
 * }

Continued Fraction Expansion of e
The constant $$ e \ $$ is mathematics is commonly known as know as Euler's number, so called due mathmetician Leonard Euler's use of the term $$ e \ $$ in his work titled Mechanica published in 1736. It is also sometime called Napier's constant for John Napier who introduced $$ e \ $$ as the base of the natural logarithm function. The constant $$ e \ $$ was first intoduced in Napier's work, but was not called $$ e \ $$ at that time. It was not until 1748 when Euler published Introductio in Analysin infinitorum that he gave a full treatment of the ideas surrounding the term $$ e \ $$. In this writing, Euler showed that: $$ e = a+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots \ $$ and that $$ e \ $$ is the limit of $$ \left ( 1+\frac{1}{1} \right )^n \ $$ as n tends towards infinity. When the term $$ e^x \ $$ is brought up in mathematics today, it has several common ways in which it is defined. The first use is as the base 10 logarithm of $$ e \ $$. Many will see the function $$ e^x \ $$ and believe it is referring to the exponential function. The term $$ e \ $$ is an irrational transcendental number, meaning it is not the root of a non-zero polynomial with rational coefficients. The number $$ e \ $$ is roughly equivalent to 2.71828, which is the base of the natural logarithm. The term $$ e \ $$ is arguably the second most important term in mathematics, behind only $$ \pi \ $$. $$ e \ $$ is defined as: $$ e= \lim_{x \to \infty}\left ( 1+\frac{1}{x} \right )^x \ $$ In 1737, Euler proved that $$ e \ $$ is irrational by showing that it has an infinite simple continued fraction: $$ e= \left [ 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, \cdots \right ]  \ $$ Euler also gave us two forms of the continued fraction expansion of $$ e \ $$ with the first as: $$ \frac{e-1}{2} = \cfrac{1}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots}}}}} $$ and the second continued fraction expansion with the noted pattern of (1,1,2), (1,1,2n) , (1,1,3n)...etc as n tends to infinity thus showing that the pattern will never repeat or converge to a final solution. $$ e-1 = 1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\ddots}}}}}}}} $$ which can also be shown in compact form as: $$ e =\lim_{n \to \infty}[2;1,\mathbf 2,1,1,\mathbf 4,1,1,\mathbf 6,1,1,\mathbf 8,1,1,...,\mathbf {2n},1,1] = [1;\mathbf 0,1,1,\mathbf 2,1,1,\mathbf 4,1,1,...] $$ In contrast to Euler's wonderful formulas,the beautiful, non-simple continued fraction expansion of $$ e \ $$ is given by: $$ e = 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\ddots}}}}} $$ As with $$ \pi \ $$ the author is unable to proof the irrational transcendental number $$ e \ $$ as there is no starting point, just a calculation that would take one to a certain number of decimal places. Euler was able to derive this using a set of formula's given here: How Euler Did It

Trigonometry
Ttrigonometric functions arose from the connection between mathematics and astronomy. The first works of trigonometric related functions were for the chords of a circle. Given a circle of fixed radius, 60 units were often used in early calculations, then the problem was to find the length of the chord subtended by a given angle. For a circle of unit radius the length of the chord subtended by the angle x was: $$ 2 \sin \frac{x}{2} \ $$. The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 B.C.E.. Although these tables did not survive, it is claimed that twelve books of tables of chords were written by Hipparchus. Hipparchus is assumed to be the founder of trigonometry due to theses volumes of tables of chords. The first appearance of the term "sine" of an angle appears in the Hindus work. Aryabhata, in about 500 C.E., gave tables of half chords which now really are sine tables and used the term "jya" for our sin. This same table was reproduced in the work of Brahmagupta (in 628) and a detailed method for constructing a table of sines for any angle were give by Bhaskara in 1150. The Arabs worked with sines and cosines and by 980 Abu'l-Wafa knew that $$ \sin {2x} = 2 \sin x \cos x \ $$ The Hindu word "jya" for the sine was adopted by the Arabs who called the sine "jiba", a meaningless word with the same sound as jya. Now jiba became jaib in later Arab writings and this word does have a meaning, namely a 'fold'. When European authors translated the Arabic mathematical works into Latin they translated jaib into the word sinus meaning fold in Latin. In particular Fibonacci's use of the term "sinus rectus arcus" soon encouraged the universal use of sine. The term sine certainly was not accepted straight away as the standard notation by all authors. In times when mathematical notation was in itself a new idea many used their own notation. Edmund Gunter was the first to use the abbreviation sin in 1624 in a drawing. The first use of sin in a book was in 1634 by the French mathematician Hérigone. The tangent term came from a different path than the sine function. The term "Tangent" developed and was not at first associated with angles. It became important for calculating heights from the length of the shadow that the object cast. The length of shadows was also of importance in the sundial. Thales used the lengths of shadows to calculate the heights of pyramids. The first known tables of shadows were produced by the Arabs around 860 and used two measures translated into Latin as "umbra recta". The name tangent was first used by Thomas Fincke in 1583. Abbreviations for tan followed a similar development to those of the sin and cos. The first occurrence of the abbreviation of "tan" was used by Albert Girard in 1626, but tan was written over the angle.

Continued Fraction Expansion of sin x
The function sin x may be defined in several different manners. The first treatment is usually associated with trigonometry, where sin x is defined as: Given a, b and c as the leg lengths of a triangle with opposite angles defined as A, B and C as shown below: the law of sines is defined as: $$ \frac{a}{ \sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2*R \ $$, where $$ R \ $$ is equal the radius of the circle.

The sine function can also be represented as a generalized continued fraction:

$$ \sin x = \cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 + \cfrac{2\cdot3 x^2}{4\cdot5-x^2 + \cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}} \ $$

The continued fraction representation expresses the real number values, both rational and irrational, of the sine function. Due to minimal amount of material on this topic, the author was unable to expand upon the given continued fraction for $$ \sin x \ $$.

Continued Fraction Expansion of tan x
The tangent function was mentioned in 1583 by T. Fincke who introduced the word "tangens" in Latin. E. Gunter (1624) used the notation "tan", and J. H. Lambert (1770) discovered the continued fraction representation of this function. The most common use in methamatics, particularly trigonometry for the function tan (x) is given by: $$ \tan x = \frac{\sin x}{\cos x} $$ Where, referencing the triangle, as shown in the section Continued Fraction Expansion of sin x $$ \tan A = \frac{a}{c} $$ Another common use for tangency is when a line or plane touches a given curve or solid at a single point. This use of tangency is commonly used in Computer Aided Drafting (CAD). These geometrical objects are called tangent lines or planes. The following continued fraction is known as Gauss's Continued Fraction, so named after the famous mathemetician Carl Friedrich Gauss. Althought Gauss gave this form of the continued fraction, he did not give a proof.

Let $$f_0, f_1, f_2, \dots$$ be a sequence of analytic functions so that
 * $$f_{i-1} - f_i = k_i\,z\,f_{i+1}$$

for all $$i > 0$$, where each $$k_i$$ is a constant.

Then
 * $$\frac{f_{i-1}}{f_i} = 1 + k_i z \frac{f_{i+1}},\, \frac{f_i}{f_{i-1}} = \frac{1}{1 + k_i z \frac{f_{i+1}}}$$.

Setting $$g_i = f_i / f_{i-1}$$,
 * $$g_i = \frac{1}{1 + k_i z g_{i+1}}$$,

So
 * $$g_1 = \frac{f_1}{f_0} = \cfrac{1}{1 + k_1 z g_2} = \cfrac{1}{1 + \cfrac{k_1 z}{1 + k_2 z g_3}}

= \cfrac{1}{1 + \cfrac{k_1 z}{1 + \cfrac{k_2 z}{1 + k_3 z g_4}}} = \dots\ $$.

Repeating this ad infinitum produces the continued fraction expression
 * $$\frac{f_1}{f_0} = \cfrac{1}{1 + \cfrac{k_1 z}{1 + \cfrac{k_2 z}{1 + \cfrac{k_3 z}{1 + {}\ddots}}}}$$

In Gauss's continued fraction, the functions $$f_i$$ are hypergeometric functions of the form $${}_0F_1$$, $${}_1F_1$$, and $${}_2F_1$$, and the equations $$f_{i-1} - f_i = k_i z f_{i+1}$$ arise as identities between functions where the parameters differ by integer amounts. These identities can be proved in several ways, for example by expanding out the series and comparing coefficients, or by taking the derivative in several ways and eliminating it from the equations generated. The simplest case involves the following: $$\,_0F_1(a;z) = 1 + \frac{1}{a\,1!}z + \frac{1}{a(a+1)\,2!}z^2 + \frac{1}{a(a+1)(a+2)\,3!}z^3 + \cdots\ $$. Starting with the identity $$\,_0F_1(a-1;z)-\,_0F_1(a;z) = \frac{z}{a(a-1)}\,_0F_1(a+1;z)$$, we may take
 * $$f_i = {}_0F_1(a+i;z),\,k_i = \tfrac{1}{(a+i)(a+i-1)}$$,

giving $$\frac{\,_0F_1(a+1;z)}{\,_0F_1(a;z)} = \cfrac{1}{1 + \cfrac{\frac{1}{a(a+1)}z} {1 + \cfrac{\frac{1}{(a+1)(a+2)}z}{1 + \cfrac{\frac{1}{(a+2)(a+3)}z}{1 + {}\ddots}}}}$$ or $$\frac{\,_0F_1(a+1;z)}{a\,_0F_1(a;z)} = \cfrac{1}{a + \cfrac{z} {(a+1) + \cfrac{z}{(a+2) + \cfrac{z}{(a+3) + {}\ddots}}}}$$.

We have

$$\cosh(z) = \,_0F_1({\tfrac{1}{2}};{\tfrac{z^2}{4}}),$$ $$\sinh(z) = z\,_0F_1({\tfrac{3}{2}};{\tfrac{z^2}{4}}),$$

so

$$\tanh(z) = \frac{z\,_0F_1({\tfrac{3}{2}};{\tfrac{z^2}{4}})}{\,_0F_1({\tfrac{1}{2}};{\tfrac{z^2}{4}})} = \cfrac{z/2}{\tfrac{1}{2} + \cfrac{\tfrac{z^2}{4}}{\tfrac{3}{2} + \cfrac{\tfrac{z^2}{4}}{\tfrac{5}{2} + \cfrac{\tfrac{z^2}{4}}{\tfrac{7}{2} + {}\ddots}}}} = \cfrac{z}{1 + \cfrac{z^2}{3 + \cfrac{z^2}{5 + \cfrac{z^2}{7 + {}\ddots}}}}.$$

This particular expansion is known as Lambert's continued fraction and dates back to 1768.

It easily follows that

$$\tan(z) = \cfrac{z}{1 - \cfrac{z^2}{3 - \cfrac{z^2}{5 - \cfrac{z^2}{7 - {}\ddots}}}}.$$

The expansion of tanh can be used to prove that en is irrational for every integer n (which is alas not enough to prove that e is transcendental). The expansion of tan was used by both Lambert and Legendre]] to prove that π is irrational.

This expansion converges provided, of course, that a is neither zero nor a negative integer.