List of trigonometric identities

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

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 technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Pythagorean identities

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Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity , and the red triangle shows that .

The basic relationship between the sine and cosine is given by the Pythagorean identity:

where means and means

This can be viewed as a version of the Pythagorean theorem, and follows from the equation for the unit circle. This equation can be solved for either the sine or the cosine:

where the sign depends on the quadrant of

Dividing this identity by , , or both yields the following identities:

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

Each trigonometric function in terms of each of the other five.[1]
in terms of


Reflections, shifts, and periodicity

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By examining the unit circle, one can establish the following properties of the trigonometric functions.

Reflections

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Unit circle with a swept angle theta plotted at coordinates (a,b). As the angle is reflected in increments of one-quarter pi (45 degrees), the coordinates are transformed. For a transformation of one-quarter pi (45 degrees, or 90 – theta), the coordinates are transformed to (b,a). Another increment of the angle of reflection by one-quarter pi (90 degrees total, or 180 – theta) transforms the coordinates to (-a,b). A third increment of the angle of reflection by another one-quarter pi (135 degrees total, or 270 – theta) transforms the coordinates to (-b,-a). A final increment of one-quarter pi (180 degrees total, or 360 – theta) transforms the coordinates to (a,-b).
Transformation of coordinates (a,b) when shifting the reflection angle in increments of .

When the direction of a Euclidean vector is represented by an angle this is the angle determined by the free vector (starting at the origin) and the positive -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive -axis. If a line (vector) with direction is reflected about a line with direction then the direction angle of this reflected line (vector) has the value

The values of the trigonometric functions of these angles for specific angles satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.[2]

reflected in [3]
odd/even identities
reflected in reflected in reflected in reflected in
compare to

Shifts and periodicity

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Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).
Transformation of coordinates (a,b) when shifting the angle in increments of .
Shift by one quarter period Shift by one half period Shift by full periods[4] Period

Signs

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The sign of trigonometric functions depends on quadrant of the angle. If and sgn is the sign function,

The trigonometric functions are periodic with common period so for values of θ outside the interval they take repeating values (see § Shifts and periodicity above).

Angle sum and difference identities

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Illustration of angle addition formulae for the sine and cosine of acute angles. Emphasized segment is of unit length.
Diagram showing the angle difference identities for and .

These are also known as the angle addition and subtraction theorems (or formulae).

The angle difference identities for and can be derived from the angle sum versions by substituting for and using the facts that and . They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.

These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.

Sine [5][6]
Cosine [6][7]
Tangent [6][8]
Cosecant [9]
Secant [9]
Cotangent [6][10]
Arcsine [11]
Arccosine [12]
Arctangent [13]
Arccotangent

Sines and cosines of sums of infinitely many angles

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When the series converges absolutely then

Because the series converges absolutely, it is necessarily the case that and In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.

Tangents and cotangents of sums

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Let (for ) be the kth-degree elementary symmetric polynomial in the variables for that is,

Then

using the sine and cosine sum formulae above.

The number of terms on the right side depends on the number of terms on the left side.

For example:

and so on. The case of only finitely many terms can be proved by mathematical induction.[14] The case of infinitely many terms can be proved by using some elementary inequalities.[15]

Secants and cosecants of sums

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where is the kth-degree elementary symmetric polynomial in the n variables and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[16] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

Ptolemy's theorem

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Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.[17] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.

By Thales's theorem, and are both right angles. The right-angled triangles and both share the hypotenuse of length 1. Thus, the side , , and .

By the inscribed angle theorem, the central angle subtended by the chord at the circle's center is twice the angle , i.e. . Therefore, the symmetrical pair of red triangles each has the angle at the center. Each of these triangles has a hypotenuse of length , so the length of is , i.e. simply . The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also .

When these values are substituted into the statement of Ptolemy's theorem that , this yields the angle sum trigonometric identity for sine: . The angle difference formula for can be similarly derived by letting the side serve as a diameter instead of .[17]

Multiple-angle and half-angle formulae

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Tn is the nth Chebyshev polynomial [18]
de Moivre's formula, i is the imaginary unit [19]

Multiple-angle formulae

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Double-angle formulae

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Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle , the area 1/2 × base × height is calculated in two orientations. When upright, the area is . When on its side, the same area is . Therefore,

Formulae for twice an angle.[20]

Triple-angle formulae

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Formulae for triple angles.[20]

Multiple-angle formulae

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Formulae for multiple angles.[21]

Chebyshev method

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The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the th and th values.[22]

can be computed from , , and with

This can be proved by adding together the formulae

It follows by induction that is a polynomial of the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

Similarly, can be computed from and with This can be proved by adding formulae for and

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

Half-angle formulae

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[23][24]

Also