Friday 8 March 2013

Trigonometry

The Canadarm2 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.

Trigonometry

History
Usage
Functions
Generalized
Inverse functions
Further reading

Reference

Identities
Exact constants
Trigonometric tables

Laws and theorems

Law of sines
Law of cosines
Law of tangents
Law of cotangents
Pythagorean theorem

Calculus

Trigonometry (from Greek trigōnon "triangle" + metron "measure"^[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies. It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.

In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$

Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:

$\csc A=\frac{1}{\sin A}=\frac{c}{a} ,$

$\sec A=\frac{1}{\cos A}=\frac{c}{b} ,$

$\cot A=\frac{1}{\tan A}=\frac{\cos A}{\sin A}=\frac{b}{a} .$

The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".
With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

Extending the definitions

Fig. 1a – Sine and cosine of an angle θ defined using the unit circle.

The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.

$e^{x+iy} = e^x(\cos y + i \sin y).$

See Euler's and De Moivre's formulas.

Graphing process of y = sin(x) using a unit circle.
Graphing process of y = tan(x) using a unit circle.
Graphing process of y = csc(x) using a unit circle.

Mnemonics

Mnemonics in trigonometry

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters. For instance, a mnemonic for English speakers is SOH-CAH-TOA:

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-kə-tow'-uh').^[14] Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".^[15]

Calculating trigonometric functions

Generating trigonometric tables Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad.^{[citation needed]} Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.

Applications of trigonometry

Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.

Main article: Uses of trigonometry

There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

Standard identities

Identities are those equations that hold true for any value.

$\sin^2 A + \cos^2 A = 1 \$

$\sec^2 A - \tan^2 A = 1 \$

$\csc^2 A - \cot^2 A = 1 \$

Angle transformation formulas

$\sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B$

$\cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B$

$\tan (A \pm B) = \frac{ \tan A \pm \tan B }{ 1 \mp \tan A \ \tan B}$

$\cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A }$

Common formulas

Triangle with sides a,b,c and respectively opposite angles A,B,C

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles.

Law of sines

The law of sines (also known as the "sine rule") for an arbitrary triangle states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,$

where R is the radius of the circumscribed circle of the triangle:

$R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.$

Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:

$\mbox{Area} = \frac{1}{2}a b\sin C.$

All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

Law of cosines

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:

$c^2=a^2+b^2-2ab\cos C ,\,$

or equivalently:

$\cos C=\frac{a^2+b^2-c^2}{2ab}.\,$

Law of tangents

The law of tangents:

$\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}$

Euler's formula

Euler's formula, which states that $e^{ix} = \cos x + i \sin x$ , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:

$\sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.$

ALGEBRAIC EQUATION

Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form

$P = Q$

where P and Q are polynomials with coefficients in some field, often the field of the rational numbers. For most authors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate and the term polynomial equation is usually preferred to algebraic equation.
For example,

$x^5-3x+1=0$

is an algebraic equation with integer coefficients and

$y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}$

is a polynomial equation over the rationals.
The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kind of quadratic equations (displayed on Old Babylonian clay tablets).
Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with a finite number of operations involving just those coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations but not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of an univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).
The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals. Galois theory has been introduced by Évariste Galois for getting criteria deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element is a root of an algebraic equation over the base field. Transcendence theory is the study of the real numbers which are not solutions to an algebraic equation over the rationals. A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations.
Two equations are equivalent if they have the same set of solutions. In particular the equation $P = Q$ is equivalent with $P-Q = 0$ . It follows that the study of algebraic equations is equivalent to the study of polynomials.
A polynomial equation over the rationals can always be converted to an equivalent one in which the coefficients are integers. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the polynomial equation above becomes

$42y^4+21xy-14x^3+42xy^2-42y^2+6=0.$

Because sine, exponentiation, and 1/T are not polynomial functions,

$e^T x^2+\frac{1}{T}xy+\sin(T)z -2 =0$

is not a polynomial equation in the four variables x, y, z, and T over the rational numbers. However, it is a polynomial equation in the three variables x, y, and z over the field of the elementary functions in the variable T.
As for any equation, the solutions of an equation are the values of the variables for which the equation is true. For univariate algebraic equations these are also called roots, even if, properly speaking, one should say the solutions of the algebraic equation P=0 are the roots of the polynomial P. When solving an equation, it is important to specify in which set the solutions are allowed. For example, for an equation over the rationals one may look for solutions in which all the variables are integers. In this case the equation is a diophantine equation. One may also be interested only in the real solutions. However, for univariate algebraic equations, the number of solutions is finite and all solutions, are contained in any algebraically closed field containing the coefficients, for example, the field of complex numbers in case of equations over the rationals. It follows that without precision "root" and "solution" usually mean "solution in an algebraically closed field".
The algebraic equations over the rationals with only one variable are also called univariate equations. They have a very long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like $x=\frac{1+\sqrt{5}}{2}$ for the positive solution of $x^2+x-1=0$ . The ancient Egyptians knew how to solve equations of degree 2 in this manner. During the Renaissance, Gerolamo Cardano has found the solution of the equation of degree 3 and Lodovico Ferrari solved the equation of degree 4. Finally Niels Henrik Abel proved, in 1824, that the equation of degree 5 and equations of higher degree are not always solvable using radicals. Galois theory, named after Évariste Galois, were introduced to give criteria deciding if an equation is solvable using radicals.

Sunday 17 February 2013

MATHEMATICS MADE EASY

Algebraic expression

an algebraic expression is an expression built up from constants, variables, and a finite number of algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).For example, $3x^2 - 2xy + c$ is an algebraic expression. Since taking the square root is the same as raising to the power $\tfrac{1}{2}$ ,

$\sqrt{\frac{1-x^2}{1+x^2}}$

is also an algebraic expression.
Italic textterminology algebrac is an entegral expression that all non entegers and none negative numbers

1 – Exponent (power), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, $x, y$ - variables

Variables

By convention, letters at the beginning of the alphabet (e.g. $a, b, c$ ) are typically used to represent constants, and those toward the end of the alphabet (e.g. $x, y$ and $z$ ) are used to represent variables. They are usually written in italics.

Exponents

By convention, terms with the highest power (exponent), are written on the left, for example, $x^2$ is written to the left of $x$ . When a coefficient is one, it is usually omitted (e.g. $1x^2$ is written $x^2$ ). Likewise when the exponent (power) is one, (e.g. $3x^1$ is written $3x$ ), and, when the exponent is zero, the result is always 1 (e.g. $x^0$ is always $1$ ).

Rational expressions

A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as

x 2 + 2 x + 4

. An irrational algebraic expression is one that is not rational, such as

\sqrt x + 4