Trigonometry

Trigonometry ( ) is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.

Trigonometry ( ) is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. \sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{h}. Cosine (denoted cos), defined as the ratio of the adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse. \cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{h}. Tangent (denoted tan), defined as the ratio of the opposite leg to the adjacent leg. \tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{b}=\frac{a/h}{b/h}=\frac{\sin A}{\cos A}. The reciprocals of these ratios are named the cosecant (csc), secant (sec), and cotangent (cot), respectively: \csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{h}{a} , \sec A=\frac{1}{\cos A}=\frac{\textrm{hypotenuse}}{\textrm{adjacent}}=\frac{h}{b} , \cot A=\frac{1}{\tan A}=\frac{\textrm{adjacent}}{\textrm{opposite}}=\frac{\cos A}{\sin A}=\frac{b}{a} . 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. === Mnemonics === 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 and their corresponding sides as strings of letters. For instance, a mnemonic 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. , similar to Krakatoa). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid". ===The unit circle and common trigonometric values=== Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane. In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where x = \cos A and y = \sin A . This representation allows for the calculation of commonly found trigonometric values, such as those in the following table: \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots &&= \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \\ \cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots &&= \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} \end{align} With these definitions the trigonometric functions can be defined for complex numbers. When extended as functions of real or complex variables, the following formula holds for the complex exponential: e^{x+iy} = e^x(\cos y + i \sin y). This complex exponential function, written in terms of trigonometric functions, is particularly useful. ===Astronomy=== For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, predicting eclipses, and describing the orbits of the planets. In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, as well as in satellite navigation systems. ===Navigation=== s 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.]] Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation. Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles. ===Surveying=== In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects. On a larger scale, trigonometry is used in geography to measure distances between landmarks. ===Periodic functions=== vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).]] The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions. Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics and communications, among other fields. ===Optics and acoustics=== Trigonometry is useful in many physical sciences, including acoustics, and optics. In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems. ===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 (as shown in the diagram). \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R = \frac{abc}{2\Delta}, where \Delta is the area of the triangle and 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)}}. c^2=a^2+b^2-2ab\cos C , or equivalently: \cos C=\frac{a^2+b^2-c^2}{2ab}. \frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]} \mbox{Area} = \Delta = \frac{1}{2}a b\sin C ===Trigonometric identities=== \sin^2 A + \cos^2 A = 1 \ \tan^2 A + 1 = \sec^2 A \ \cot^2 A + 1 = \csc^2 A \ The second and third equations are derived from dividing the first equation by \cos^2{A} and \sin^2{A} , respectively. 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}}. ==Further reading== Linton, Christopher M. (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press. Benjamin Banneker's Trigonometry Puzzle at Convergence Dave's Short Course in Trigonometry by David Joyce of Clark University Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License

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