What is sin 2 x
The functions sin x and cos x can be expressed by series that converge for all values of x: These series can be used to obtain approximate expressions for sin x and cos x for small values of x: The trigonometric system 1, cos x, sin x, cos 2 x, sin 2 x,, cos nx, sin nx, constitutes an orthogonal system of functions on the. [math]sin(2x) = 2sinxcosx[/math] This can be derived by another Trigonometric Function, [math]sin(2x) = sin(x+x)[/math] [math]sin(A+B) = sinAcosB + cosAsinB[/math.
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Or would it be equal to one-quarter, or would it be sni to something completely different? For definition of a radian and other details, you may want to look up Wikipedia.
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One source is: mathworld. Add a comment. Active Oldest Votes. Improve this answer. The smallest positive solution appears to be about 1. I was not being careful Tanner Swett Tanner Swett 8, 25 25 silver how to recover password in iphone 49 49 bronze badges.
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Jan 25, · sin 2 x MEANS (sin x) 2. Which you can calculate on your likedatingall.comted Reading Time: 30 secs. sin^2(x) Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music Wolfram|Alpha brings expert-level knowledge and. The usual convention is that sin2(X) = (sin(X))2. So for your example 1 / 4 is correct. sin(30 ?) = 1 / 2, thus sin2(30 ?) = (sin(30 ?))2 = (1 / 2)2 = 1 / 4. However, sin( ?) = sin( ? ? 5) = sin( ?) = 0 because sin( ? ? k) = 0 for any integer k.
Figure 1. Arcs of arbitrary length are plotted from point A along the perimeter. Since the radius of the circle is unity, the central angle in radian measurement is measured by the same number as the arc. In general, the argument of a trigonometric function is considered to be a number, which may be regarded geometrically as the length of an arc or the radian measure of an angle.
The most important such functions are the tangent tan , cotangent cot, or ctn , secant sec , and cosecant csc :. The Hindu mathematicians had used the term jiva , which means bowstring, to refer to the sine. Thus, we have. Like the sine and cosine, the other trigonometric functions of acute angles may be regarded as the ratios of the lengths of the sides of a right triangle. The tangent is the ratio of the opposite leg and the adjacent leg, the cotangent is the ratio of the adjacent leg and the opposite leg, the secant is the ratio of the hypotenuse and the adjacent leg, and the cosecant is the ratio of the hypotenuse and the opposite leg.
Consequently, all trigonometric functions are periodic. The graphs of these trigonometric functions are illustrated in Figure 2. Figure 2. Graphs of trigonometric functions: 1 sine, 2 cosine, 3 tangent, 4 cotangent, 5 secant, 6 cosecant. The following relations, sometimes called the Pythagorean. For some values of the argument, the values of the trigonometric functions can be obtained from geometric considerations see Table 1.
For large values of the argument, identities called reduction formulas may be used. This circumstance simplifies the construction and use of tables of trigonometric functions and the. Another important group of identities consists of addition formulas, which express the trigonometric functions of the sum or difference of the values of the argument in terms of the trigonometric functions of these values:. In each formula either the upper signs or the lower signs are to be used throughout; that is, the upper sign on the right side of the formula corresponds to the upper sign on the left side, and the lower sign on the right side corresponds to the lower sign on the left side.
Formulas for the trigonometric functions of multiple arguments can be derived from the addition formulas—for example,. The above identities are often called double-angle formulas.
Formulas that express the powers of the sine and cosine of an argument in terms of the sine and cosine of multiples of the argument are frequently useful. Examples are. Equations 3 are called half-angle formulas. The sums or differences of trigonometric functions of different arguments can be converted into products according to the following formulas:. Here, in the first formula and in the last formula either the upper signs or the lower signs are used throughout.
The products of trigonometric functions can be converted into a sum or difference by means of the formulas. The derivatives of the trigonometric functions can be expressed in terms of trigonometric functions:. The integrals of the trigonometric functions are trigonometric functions or logarithms of trigonometric functions:. The integrals of rational combinations of trigonometric functions are always elementary functions.
All trigonometric functions can be expanded in power series. The functions sin x and cos x can be expressed by series that converge for all values of x:. These series can be used to obtain approximate expressions for sin x and cos x for small values of x :. The trigonometric system 1, cos x , sin x , cos 2 x , sin 2 x ,. This fact makes it possible to represent functions by trigonometric series. For complex values of the argument, the values of trigonometric functions can be determined by means of power series.
From this formula expressions can be obtained for sin x and cos x in terms of the exponential functions of a purely imaginary argument:. The sine and cosine of a complex argument may assume real values that exceed 1 in absolute value. For example,. The trigonometric functions of a complex argument are analytic functions. Furthermore, sin z and cos z are entire functions, and tan z, cot z, sec z, and csc z are meromorphic functions.
The strip , which is half as wide, is mapped into the upper half plane. This function is the inverse of the sine and is symbolized Arc sin x. The inverse functions of the cosine, tangent, cotangent, secant, and cosecant are defined in a similar way; they are Arc cos x , Arc tan x , Arc cot x , Arc sec x , and Arc csc x. Such functions are called inverse trigonometric functions.
In non-Russian mathematical literature they are sometimes denoted sin —1 z, cos —1 z, and so on. Trigonometric functions were first used in connection with studies in astronomy and geometry. The relations among line segments in a triangle and circle are in essence trigonometric functions, and these relations were employed as early as the third century B.
The relations, however, did not constitute an independent subject of study for these mathematicians; consequently, it cannot be said that trigonometric functions as such were investigated by them. Trigonometric functions were originally treated as line segments and were used in this form in the solution of spherical triangles by Aristarchus late fourth and second half of third centuries B. Formulas 4 were derived by Regiomontanus 15th century and by J.
Napier in connection with his invention of logarithms Newton obtained expansions of trigonometric functions in power series The theory of trigonometric functions was given its modern form by L.
Euler 18th century. He defined trigonometric functions for real and complex arguments, provided the notation now used, established the relation between the exponential function and trigonometric functions, and showed the orthogonality of the system of sines and cosines.
The following article is from The Great Soviet Encyclopedia It might be outdated or ideologically biased. Trigonometric functions make up one of the most important classes of elementary functions. Algebra i elementarnye funktsii , parts 1—2. Moscow, Shabat, B. Vvedenie v kompleksnyi onaliz.
Pages 61— The Great Soviet Encyclopedia, 3rd Edition All rights reserved. Mentioned in? Encyclopedia browser? Full browser?
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