trigonometry

trigonometry

[trig-uh-nom-i-tree]
trigonometry [Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy.

The Basic Trigonometric Functions

Trigomometry originated as the study of certain mathematical relations originally defined in terms of the angles and sides of a right triangle, i.e., one containing a right angle (90°). Six basic relations, or trigonometric functions, are defined.If A, B, and C are the measures of the angles of a right triangle (C=90°) and a, b, and c are the lengths of the respective sides opposite these angles, then the six functions are expressed for one of the acute angles, say A, as various ratios of the opposite side (a), the adjacent side (b), and the hypotenuse (c), as set out in the table.Although the actual lengths of the sides of a right triangle may have any values, the ratios of the lengths will be the same for all similar right triangles, large or small; these ratios depend only on the angles and not on the actual lengths. The functions occur in pairs—sine and cosine, tangent and cotangent, secant and cosecant—called cofunctions. In equations they are usually represented as sin, cos, tan, cot, sec, and csc. Since in ordinary (Euclidean) plane geometry the sum of the angles of a triangle is 180°, angles A and B must add up to 90° and therefore are complementary angles. From the definitions of the functions, it may be seen that sin B=cos A, cos B=sin A, tan B=cot A, and sec B=csc A; in general, the function of an angle is equal to the cofunction of its complement. Since the hypotenuse (c), is always the longest side of a right triangle, the values of the sine and cosine are always between zero and one, the values of the secant and cosecant are always equal to or greater than one, and the values of the tangent and cotangent are unbounded, increasing from zero without limit.

For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 :3 : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°=3/2, tan 30°=cot 60°=1/3, cot 30°=tan 60°=3, sec 30°=csc 60°=2/3, and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.

A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a/sin A=b/sin B=c/sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a2=b2+c2-2bc cosA and the Law of Tangents holds that (a-b)/(a+b)=[tan 1/2(A-B)]/[tan 1/2(A+B)]. Each of the trigonometric functions can be represented by an infinite series.

Extension of the Trigonometric Functions

The notion of the trigonometric functions can be extended beyond 90° by defining the functions with respect to Cartesian coordinates. Let r be a line of unit length from the origin to the point P (x,y), and let θ be the angle r makes with the positive x-axis. The six functions become sin θ =y/r=y, cos θ=x/r=x, tan θ=y/x, cot θ=x/y, sec θ=r/x=1/x, and csc θ=r/y=1/y. As θ increases beyond 90°, the point P crosses the y-axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270° P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive.Since the positions of r for angles of 360° or more coincide with those already taken by r as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on.This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.

Mathematical discipline dealing with the relationships between the sides and angles of triangles. Literally, it means triangle measurement, though its applications extend far beyond geometry. It emerged as a rigorous discipline in the 15th century, when the demand for accurate surveying techniques and navigational methods led to its use for the “solution” of right triangles, or the calculation of the lengths of two sides of a right triangle given one of its acute angles and the length of one side. The solution can be found by using ratios in the form of the trigonometric functions.

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Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called "trig".

A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.

History

Trigonometry was probably developed for use in sailing as a navigation method used with astronomy. The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley (India), more than 4000 years ago. The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration.

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD.

The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.

The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.

In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical geometry:

frac{sin A}{sin a} = frac{sin B}{sin b} = frac{sin C}{sin c}.

Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula

cos a cos b = frac{cos(a+b) + cos(a-b)}{2}.

Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation x^3 + 200 x = 20 x^2 + 2000 and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables.

Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry.

The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.

In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy.

The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry" itself.

Overview

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 right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. 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:

  • The 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,},.

  • The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

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

  • The 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 people 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. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

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 an 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

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 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 function cis is particularly useful

operatorname{cis},x = cos x + isin x ! = e^{ix}.

See Euler's and De Moivre's formulas.

Mnemonics

Students often use mnemonics 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, as in SOH-CAH-TOA.

Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent

Alternatively, one can devise sentences which consist of words beginning with the letters to be remembered. For example, to recall that Tan = Opposite/Adjacent, the letters T-O-A must be remembered. Any memorable phrase constructed of words beginning with the letters T-O-A will serve.

It is of ethnographic interest to note that the mnemonic TOA-CAH-SOH can be translated in the local Singaporean Hokkien dialect to 'big-legged woman', serving as an additional learning aid for students in Singapore.

Another type of mnemonic describes facts in a simple, memorable way, such as "Plus to the right, minus to the left; positive height, negative depth," which refers to trigonometric functions generated by a revolving line.

Calculating trigonometric functions

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. 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

There are an enormous number of applications 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 which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the 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.

Common formulae

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships. For example, the Pythagorean identities are an expression of the Pythagorean Theorem. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see trigonometric identity.

Trigonometric identities

Pythagorean identities

begin{align}
sin^2 alpha + cos^2 alpha = 1 tan^2 alpha + 1 = sec^2 alpha 1+cot^2 alpha = csc^2 alpha end{align}

Sum and product identities

Sum to product
begin{align}
sin alpha pm sin beta &= 2sin left(frac{alpha pm beta}{2}right)cos left(frac{alpha mp beta}{2} right) cos alpha + cos beta &= 2cos left(frac{alpha + beta}{2} right)cos left(frac{alpha - beta}{2}right) cos alpha - cos beta &= -2sin left(frac{alpha + beta}{2} right) sin left(frac{alpha - beta}{2}right) end{align}
Product to sum
begin{align}
cos alpha ,cos beta &= frac{1}{2}[cos(alpha - beta) + cos (alpha + beta)] sin alpha ,sin beta &= frac{1}{2}[cos(alpha - beta) - cos (alpha + beta)] cos alpha ,sin beta &= frac{1}{2}[sin(alpha + beta) - sin (alpha - beta)] sin alpha ,cos beta &= frac{1}{2}[sin(alpha + beta) + sin (alpha - beta)] end{align}
Sine, cosine, and tangent of a sum

Detailed, diagramed proofs of the first two of these formulas are available
for download as a four-page PDF document at .

begin{align}
sin(alpha pm beta) &= sin alpha cos beta pm cos alpha sin beta cos(alpha pm beta) &= cos alpha cos beta mp sin alpha sin beta tan(alpha pm beta) &= frac{tan alpha pm tan beta}{1 mp tan alpha tan beta} end{align}

Half-angle identities

Note that pm is correct, it means it may be either one, depending on the value of A/2.

begin{align}
sin frac{A}{2} &= pm sqrt{frac{1-cos A}{2}} cos frac{A}{2} &= pm sqrt{frac{1+cos A}{2}} tan frac{A}{2} &= pm sqrt{frac{1-cos A}{1+cos A}} = frac {sin A}{1+cos A} = frac {1-cos A}{sin A} end{align}

Stereographic (or parametric) identities

begin{align}
sin alpha &= frac{2T}{1+T^2} cos alpha &= frac{1-T^2}{1+T^2} end{align}

where T=tan frac{alpha}{2}.

Triangle identities

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 know 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 circumcircle of the triangle.

Law of cosines

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

c^2=a^2+b^2-2abcos 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{tanleft[tfrac{1}{2}(A+B)right]}{tanleft[tfrac{1}{2}(A-B)right]}

See also

Notes

References

Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.

Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld.

External links

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