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Branch of Mathematics |
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Mathematics |
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ganithguru |
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ALGEBRA Series(special)
GEOMETRY Theoretical Geometry Constructions Applied geometry
TRIGONOMETRY Angles and measures Trigonometric Ratio Algebra of T-functions Properties of triangles Applied Trigonometry
CALCULUS Examples of different kinds of functions Graph of functions Limit of a function Continuity of a function Differential Calculus Integral Calculus Applied Calculus
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Ganithguru Trigonometry 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
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. 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:
In
this right triangle: sin A = a/c; cos A = b/c; tan
A = a/b. The sine function (sin), defined as the ratio of the side opposite
the angle to the hypotenuse.
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- 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.
Graphs of the functions sin(x) and cos(x), where the angle x is measured in 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 function cis is particularly useful
Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians.
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. 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 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 as strings of letters, as in
SOH- Sine = Opposite ÷ Hypotenuse Cosine = Adjacent ÷ Hypotenuse Tangent = Opposite ÷ Adjacent The memorization of this mnemonic can be aided by expanding it into a phrase, such as "Some Officers Have Curly Auburn Hair Till Old Age".[4] Any memorable phrase constructed of words beginning with the letters S-O-H-C-A-H-T-O-A will serve. 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. 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. Uses
of trigonometry
Marine sextants like this 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 then be determined from several such measurements. 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 ( Marine sextants like this 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 then be determined from several such measurements. Common
formulae Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. There are some identities which equate an expression to a different expression involving the same angles and these are listed in List of trigonometric identities, and then there are the triangle identities which relate the sides and angles of a given triangle and these are listed below. Triangle
identities Laws of sines and cosines Law
of sines
where R is the radius of the circumcircle of the triangle: 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. The law of sines (also known as the "sine rule") for an
arbitrary triangle states:
Another law involving sines can be used to calculate the area of a triangle. If you know two sides and the angle between the sides, the area of the triangle becomes:
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: or equivalently:
Law
of tangents The
law of tangents: More to come.. |
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I invite review and suggestions. All students can contact me at ganithguru for further guidance.
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The
cosine function (
The
tangent function (tan), defined as the ratio of the opposite leg to the
adjacent leg. 
