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There are an enormous number of applications of trigonometry. Of particular value is the technique of triangulation which is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. Other fields which make use of trigonometry include astronomy (and hence navigation, on the oceans, in aircraft, and in space), music theory, acoustics, opticsSee also list of optical topics. Optics is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. Optics explains and is illuminated by optical phenomena. The field of optics usually describes the, analysis of financial markets, electronicsElectronics is the study and use of electrical devices that operate by controlling the flow of electrons or other electrically charged particles in devices such as thermionic valves and semiconductors. The pure study of such devices is considered as a bra, probability theoryProbability theory Discrete mathematics Mathematical analysis Probability theory is the mathematical study of probability. Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occ, statisticsStatistics is the science and practice of developing human knowledge through the use of empirical data. It is based on statistical theory which is a branch of applied mathematics. Within statistical theory, randomness and uncertainty are modelled by proba, biologyBiology studies the variety of life clockwise from top-left E. coli tree fern, gazelle, Goliath beetle Biology is the science of life. It is concerned with the characteristics and behaviors of organisms, how species and individuals come into existence, an, medical imagingMedical imaging is the process by which physicians evaluate an area of the subject's body that is not normally visible. Medical imaging may be "clinical", seeking to diagnose and examine disease in specific human patients see pathology). Alternatively, it ( CAT scans and ultrasoundUltrasound is sound with a frequency greater than the upper limit of human hearing, approximately 20 kilohertz. Some animals, such as dogs, dolphins, and bats, have an upper limit that is greater than that of the human ear and thus can hear 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.
Two triangles are said to be similar if one can be obtained by uniformly expanding the other. This is the case if and only if their corresponding angles are equal, and it occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. The crucial fact about similar triangles is that the lengths of their sides are proportionate. That is, if the longest side of a triangle is twice that of the longest side of a similar triangle, say, then the shortest side will also be twice that of the shortest side of the other triangle, and the median side will be twice that of the other triangle. Also, the ratio of the longest side to the shortest in the first triangle will be the same as the ratio of the longest side to the shortest in the other triangle.
Using these facts, one defines trigonometric functions, starting with right triangles, triangles with one right angle (90 degrees or π/2 radians). The longest side in any triangle is the side opposite the largest angle. Because the sum of the angles in a triangle is 180 degrees or π radians, the largest angle in such a triangle is the right angle. The longest side in such a triangle is therefore the side opposite the right angle and is called the hypotenuse.
Pick two right triangles which share a second angle A. These triangles are necessarily similar, and the ratio of the side opposite to A to the hypotenuse will therefore be the same for the two triangles. It will be a number between 0 and 1 which depends only on A; we call it the sine of A and write it as sin(A). Similarly, one can define the cosine of A as the ratio of the side adjacent to A to the hypotenuse.
These are by far the most important trigonometric functions; other functions can be defined by taking ratios of other sides of the right triangles but they can all be expressed in terms of sine and cosine. These are the tangent, secant, cotangent, and cosecant.
The sine, cosine and tangent ratios in right triangles can be remembered by SOH CAH TOA (sine-opposite-hypotenuse cosine-adjacent-hypotenuse tangent-opposite-adjacent). See trigonometry mnemonics for other mnemonics.
So far, the trigonometric functions have been defined for angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one may extend them to all positive and negative arguments (see trigonometric function).
Once the sine and cosine functions have been tabulated (or computed by a calculator), one can answer virtually all questions about arbitrary triangles, 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.
Some mathematicians believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books. It is also very important for surveying.
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