List_of_trigonometric_identities

All of the trigonometric functions of an angle θ can

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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O . Many of these terms are no longer in common use. Name(s) Abbreviation(s) Value [2] versed sine, versine 1 cos( θ ) versed cosine, vercosine 1 + cos( θ ) coversed sine, coversine 1 sin( θ ) coversed cosine, covercosine 1 + sin( θ ) half versed sine, haversine half versed cosine, havercosine half coversed sine, hacoversine cohaversine half coversed cosine, hacovercosine cohavercosine exterior secant, exsecant sec( θ ) 1 exterior cosecant, excosecant csc( θ ) 1 chord Page 5 of 25 List of trigonometric identities - Wikipedia, the free encyclopedia 10/4/2011
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Symmetry, shifts, and periodicity By examining the unit circle, the following properties of the trigonometric functions can be established. Symmetry When the trigonometric functions are reflected from certain angles, the result is often one of the other trigonometric functions. This leads to the following identities: Shifts and periodicity By shifting the function round by certain angles, it is often possible to find different trigonometric functions that express the result more simply. Some examples of this are shown by shifting functions round by π /2, π and 2 π radians. Because the periods of these functions are either π or 2 π , there are cases where the new function is exactly the same as the old function without the shift. Angle sum and difference identities These are also known as the addition and subtraction theorems or formulæ . They were originally established by the 10th century Persian mathematician Ab ū al-Waf ā ' B ū zj ā n ī . One method of proving these identities is to apply Euler's formula. The use of the symbols and is described in the article plus-minus sign. Reflected in θ = 0 [3] Reflected in θ = π / 4 (co-function identities) [4] Reflected in θ = π / 2 Shift by π /2 Shift by π Period for tan and cot [5] Shift by 2 π Period for sin, cos, csc and sec [6] Page 6 of 25 List of trigonometric identities - Wikipedia, the free encyclopedia 10/4/2011
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Matrix form See also: matrix multiplication The sum and difference formulae for sine and cosine can be written in matrix form as: This shows that these matrices form a representation of the rotation group in the plane (technically, the special orthogonal group SO (2)), since the composition law is fulfilled: subsequent multiplications of a vector with these two matrices yields the same result as the rotation by the sum of the angles. Sines and cosines of sums of infinitely many terms Sine [7][8] Cosine [8][9] Tangent [8][10] Arcsine [11] Arccosine [12] Arctangent [13] Page 7 of 25 List of trigonometric identities - Wikipedia, the free encyclopedia 10/4/2011
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In these two identities an asymmetry appears that is not seen in the case of sums of finitely many terms: in each product, there are only finitely many sine factors and cofinitely many cosine factors.
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