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List of trigonometric identities
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In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for every single value of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Notation
To avoid the confusion caused by the ambiguity of sin−1(x), the reciprocals and inverses of trigonometric functions are often displayed as in this table. In representing the cosecant function, the longer form 'cosec' is sometimes used in place of 'csc'.
| Function | Inverse function | Reciprocal | Inverse reciprocal |
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| sine | sin | arcsine | arcsin | cosecant | csc | arccosecant | arccsc | | cosine | cos | arccosine | arccos | secant | sec | arcsecant | arcsec | | tangent | tan | arctangent | arctan | cotangent | cot | arccotangent | arccot |
Different angular measures can be appropriate in different situations. This table shows some of the more common systems.
Radians is the default angular measure and is the one you use if you use the exponential definitions. All angular measures are unitless.
| Degrees | 30 | 45 | 60 | 90 | 120 | 180 | 270 | 360 |
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| Radians | | | | | | | | |
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| Grads | 33 ? | 50 | 66 ? | 100 | 133 ? | 200 | 300 | 400 |
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Basic relationships
From the two identities above, the following table can be extrapolated. Note however that these conversion equations may not provide the correct sign (+ or −). For example, if sin ? = 1/2, the conversion in the table indicates that , though it is possible that . More information would be needed about which quadrant ? lies in to determine a single, exact answer.
Each trigonometric function in terms of the other five.| Function | sin | cos | tan | csc | sec | cot | | | | | | | | |
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Historic shorthandsRarely used today, the versine, coversine, haversine, and exsecant have been defined as below and used in navigation, for example the haversine formula was used to calculate the distance between two points on a sphere.
| Name(s) | Abbreviation(s) | Value |
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versed sine
versine | | | coversed sine
coversine | | | haversed sine
haversine |
| | hacoversed sine
hacoversine
cohaversine
havercosine |
| | | exsecant | | | | excosecant | | |
Symmetry, shifts, and periodicityBy examining the unit circle, the following properties of the trigonometric functions can be established.
SymmetryWhen the trigonometric functions are reflected from certain values of , The result is often one of the other trigonometric functions. This leads to the following identities:
| Reflected in | Reflected in
(co-function identities) | Reflected in |
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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 p/2, p and 2p radians. Because the periods of these functions are either p or 2p, there are cases where the new function is exactly the same as the old function without the shift.
| Shift by p/2 | Shift by p
Period for tan and cot | Shift by 2p
Period for sin, cos, csc and sec |
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Angle sum and difference identitiesThese are also known as the addition and subtraction theorems or formulæ.
The quickest way to prove these is Euler's formula.
Sines and cosines of sums of infinitely many terms
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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.
If only finitely many of the terms ?i are nonzero, then only finitely many of the terms on the right side will be nonzero because sine factors will vanish, and in each term, all but finitely many of the cosine factors will be unity.
Tangents of sums of finitely many terms
Let xi = tan(?i ), for i = 1, ..., n. Let ek be the kth-degree elementary symmetric polynomial in the variables xi, i = 1, ..., n, k = 0, ..., n. Then
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the number of terms depending on n.
For example:
and so on. The general case can be proved by mathematical induction.
Multiple-angle formulae
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(This function of x is the Dirichlet kernel.)
Double-, triple-, and half-angle formulaeThese can be shown by using either the sum and difference identities or the multiple-angle formulae.
>!colspan="4"| Double-angle formulae e="vertical-align:top"|
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!colspan="4"| Triple-angle formulae
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!colspan="4"| Half-angle formulae
The integral identities can be found in "list of integrals of trigonometric functions". Some generic forms are listed below.
ImplicationsThe fact that the differentiation of trigonometric functions (sine and cosine) results in linear combinations of the same two functions is of fundamental importance to many fields of mathematics, including differential equations and fourier transformations.
Exponential definitions
Miscellaneous
Dirichlet kernelThe Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity:
The convolution of any integrable function of period 2p with the Dirichlet kernel coincides with the function's nth-degree Fourier approximation. The same holds for any measure or generalized function.
Extension of half-angle formulae
If we set
then
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where eix is the same as cis(x).
This substitution of t for tan(x/2), with the consequent replacement of sin(x) by 2t/(1 + t2) and cos(x) by (1 − t2)/(1 + t2) is useful in calculus for converting rational functions in sin(x) and cos(x) to functions of t in order to find their antiderivatives. For more information see tangent half-angle formula.
See also
External links- , and for the same angles, .
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