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.
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:
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
By examining the unit circle, one can establish the following properties of the trigonometric functions.
Reflections
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]
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).
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.
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.
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
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.
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]
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a
compass and straightedge construction of
angle trisection to the algebraic problem of solving a
cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by
field theory. [citation needed]
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the
cubic equation4x3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the
discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle).
None of these solutions are reducible to a real
algebraic expression, as they use intermediate complex numbers under the
cube roots.
Power-reduction formulae
Obtained by solving the second and third versions of the cosine double-angle formula.
The product-to-sum identities[28] or
prosthaphaeresis formulae can be proven by expanding their right-hand sides using the
angle addition theorems. Historically, the first four of these were known as Werner's formulas, after
Johannes Werner who used them for astronomical calculations.[29] See
amplitude modulation for an application of the product-to-sum formulae, and
beat (acoustics) and
phase detector for applications of the sum-to-product formulae.
For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different
phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in
sinusoiddata fitting, because the measured or observed data are linearly related to the a and b unknowns of the
in-phase and quadrature components basis below, resulting in a simpler
Jacobian, compared to that of and .
Sine and cosine
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[33][34]
where and are defined as so:
given that
Arbitrary phase shift
More generally, for arbitrary phase shifts, we have
Euler's formula states that, for any real number x:[39]
where i is the
imaginary unit. Substituting −x for x gives us:
These two equations can be used to solve for cosine and sine in terms of the
exponential function. Specifically,[40][41]
These formulae are useful for proving many other trigonometric identities. For example, that
ei(θ+φ) = eiθeiφ means that
cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the
complex logarithm.
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[46]
Taking the
multiplicative inverse of both sides of the each equation above results in the equations for
The right hand side of the formula above will always be flipped.
For example, the equation for is:
while the equations for and are:
The following identities are implied by the
reflection identities. They hold whenever are in the domains of the relevant functions.
is a special case of an identity that contains one variable:
Similarly,
is a special case of an identity with :
For the case ,
For the case ,
The same cosine identity is
Similarly,
Similarly,
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are
relatively prime to (or have no
prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible
cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the
Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Many of those curious identities stem from more general facts like the following:[50]
and
Combining these gives us
If n is an odd number () we can make use of the symmetries to get
The transfer function of the
Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:
Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = Σn−1 k=1 arctan tk ∈ (π/4, 3π/4), let tn = tan(π/2 − θn) = cot θn. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1). In particular, the computed tn will be rational whenever all the t1, ..., tn−1 values are rational. With these values,
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the tk values is not within (−1, 1). Note that if t = p/q is rational, then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2 − p2, q2 + p2).
For example, for n = 3 terms,
for any a, b, c, d > 0.
An identity of Euclid
Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
Further "conditional" identities for the case α + β + γ = 180°
A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.[53] The following formulae apply to arbitrary plane triangles and follow from as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).
The
versine,
coversine,
haversine, and
exsecant were used in navigation. For example, the
haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.
When this substitution of for tan x/2 is used in
calculus, it follows that is replaced by 2t/1 + t2, is replaced by 1 − t2/1 + t2 and the differential dx is replaced by 2 dt/1 + t2. Thereby one converts rational functions of and to rational functions of in order to find their
antiderivatives.
^Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). Proceedings of the ACM-
SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC '89 (Portland US-OR, 1989-07). New York:
ACM. pp. 207–211.
doi:
10.1145/74540.74566.
ISBN0-89791-325-6.
^Eves, Howard (1990). An introduction to the history of mathematics (6th ed.). Philadelphia: Saunders College Pub. p. 309.
ISBN0-03-029558-0.
OCLC20842510.
^Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". American Journal of Physics. 21 (2): 140.
Bibcode:
1953AmJPh..21..140M.
doi:
10.1119/1.1933371.
^
abcWu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
^S. M. Abrarov, R. K. Jagpal, R. Siddiqui and B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", Mathematics, 9 (17), 2162,
arXiv:2107.01027,
doi:10.3390/math9172162{{
citation}}: CS1 maint: multiple names: authors list (
link)
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.
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:
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
By examining the unit circle, one can establish the following properties of the trigonometric functions.
Reflections
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]
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).
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.
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.
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
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.
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]
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a
compass and straightedge construction of
angle trisection to the algebraic problem of solving a
cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by
field theory. [citation needed]
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the
cubic equation4x3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the
discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle).
None of these solutions are reducible to a real
algebraic expression, as they use intermediate complex numbers under the
cube roots.
Power-reduction formulae
Obtained by solving the second and third versions of the cosine double-angle formula.
The product-to-sum identities[28] or
prosthaphaeresis formulae can be proven by expanding their right-hand sides using the
angle addition theorems. Historically, the first four of these were known as Werner's formulas, after
Johannes Werner who used them for astronomical calculations.[29] See
amplitude modulation for an application of the product-to-sum formulae, and
beat (acoustics) and
phase detector for applications of the sum-to-product formulae.
For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different
phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in
sinusoiddata fitting, because the measured or observed data are linearly related to the a and b unknowns of the
in-phase and quadrature components basis below, resulting in a simpler
Jacobian, compared to that of and .
Sine and cosine
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[33][34]
where and are defined as so:
given that
Arbitrary phase shift
More generally, for arbitrary phase shifts, we have
Euler's formula states that, for any real number x:[39]
where i is the
imaginary unit. Substituting −x for x gives us:
These two equations can be used to solve for cosine and sine in terms of the
exponential function. Specifically,[40][41]
These formulae are useful for proving many other trigonometric identities. For example, that
ei(θ+φ) = eiθeiφ means that
cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the
complex logarithm.
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[46]
Taking the
multiplicative inverse of both sides of the each equation above results in the equations for
The right hand side of the formula above will always be flipped.
For example, the equation for is:
while the equations for and are:
The following identities are implied by the
reflection identities. They hold whenever are in the domains of the relevant functions.
is a special case of an identity that contains one variable:
Similarly,
is a special case of an identity with :
For the case ,
For the case ,
The same cosine identity is
Similarly,
Similarly,
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are
relatively prime to (or have no
prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible
cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the
Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Many of those curious identities stem from more general facts like the following:[50]
and
Combining these gives us
If n is an odd number () we can make use of the symmetries to get
The transfer function of the
Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:
Generally, for numbers t1, ..., tn−1 ∈ (−1, 1) for which θn = Σn−1 k=1 arctan tk ∈ (π/4, 3π/4), let tn = tan(π/2 − θn) = cot θn. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t1, ..., tn−1 and its value will be in (−1, 1). In particular, the computed tn will be rational whenever all the t1, ..., tn−1 values are rational. With these values,
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the tk values is not within (−1, 1). Note that if t = p/q is rational, then the (2t, 1 − t2, 1 + t2) values in the above formulae are proportional to the Pythagorean triple (2pq, q2 − p2, q2 + p2).
For example, for n = 3 terms,
for any a, b, c, d > 0.
An identity of Euclid
Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
Further "conditional" identities for the case α + β + γ = 180°
A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.[53] The following formulae apply to arbitrary plane triangles and follow from as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).
The
versine,
coversine,
haversine, and
exsecant were used in navigation. For example, the
haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.
When this substitution of for tan x/2 is used in
calculus, it follows that is replaced by 2t/1 + t2, is replaced by 1 − t2/1 + t2 and the differential dx is replaced by 2 dt/1 + t2. Thereby one converts rational functions of and to rational functions of in order to find their
antiderivatives.
^Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). Proceedings of the ACM-
SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation. ISSAC '89 (Portland US-OR, 1989-07). New York:
ACM. pp. 207–211.
doi:
10.1145/74540.74566.
ISBN0-89791-325-6.
^Eves, Howard (1990). An introduction to the history of mathematics (6th ed.). Philadelphia: Saunders College Pub. p. 309.
ISBN0-03-029558-0.
OCLC20842510.
^Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". American Journal of Physics. 21 (2): 140.
Bibcode:
1953AmJPh..21..140M.
doi:
10.1119/1.1933371.
^
abcWu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189.
^S. M. Abrarov, R. K. Jagpal, R. Siddiqui and B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", Mathematics, 9 (17), 2162,
arXiv:2107.01027,
doi:10.3390/math9172162{{
citation}}: CS1 maint: multiple names: authors list (
link)