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(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)These can be derived using the trigonometricaddition formulas(25)(26)(27)(28)and(29)(30)(31)(32)

The angles (with integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of which are precisely those which produce constructible polygons. Analytic expressions for trigonometric functions with arguments of this form can be obtained using the Wolfram Language function ToRadicals, e.g., ToRadicals[Sin[Pi/17]], for values of (for , the trigonometric functions auto-evaluate in the Wolfram Language).Compass and straightedge constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ..., sides. However, Gauss showed in 1796 (when he was 19 years old) that a sufficient condition for a regular polygon on sides to be constructible was that be of the form(1)where is a nonnegative integer and the are distinct Fermat primes. Here, a Fermat prime is a prime Fermat number, i.e., a prime number of the..

Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 13 is not a Fermat prime. This also means that the tridecagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived using the multiple-angle formula(1)where is a Chebyshev polynomial of the first kind. Plugging in gives(2)Letting and then gives(3)But this is a sextic equation has a cyclic Galois group, and so , and hence , can be expressed in terms of radicals (of complex numbers). The explicit expression is quite complicated, but can be generated in the Wolfram Language using Developer`TrigToRadicals[Sin[Pi/13]].The trigonometric functions of can be given explicitly as the polynomial roots(4)(5)(6)(7)(8)(9)From one of the Newton-Girard formulas,(10)The trigonometric functions of also obey the identities(11)(12)(P. Rolli,..

The Werner formulas are the trigonometric product formulas(1)(2)(3)(4)This form of trigonometric functions can be obtained in the WolframLanguage using the command TrigReduce[expr].

Power formulas include(1)(2)(3)and(4)(5)(6)(Beyer 1987, p. 140). Formulas of these types can also be given analytically as(7)(8)(9)(10)where is a binomial coefficient.Additional useful power identities include(11)which the Wolfram Language's FullSimplify command unfortunately does not know about.

The Weierstrass substitution is the trigonometric substitution which transforms an integral of the forminto one of the formAccording to Spivak (2006, pp. 382-383), this is undoubtably the world's sneakiest substitution.The Weierstrass substitution can also be useful in computing a Gröbner basis to eliminate trigonometric functions from a system of equations (Trott 2006, p. 39).

Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 11 is not a Fermat prime. This also means that the hendecagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived using the multiple-angle formula(1)where is a Chebyshev polynomial of the first kind. Plugging in gives(2)Letting and then gives(3)But this quintic equation has a cyclic Galois group, and so , and hence , can be expressed in terms of radicals (of complex numbers). The explicit expression is quite complicated, but can be generated in the Wolfram Language using Developer`TrigToRadicals[Sin[Pi/11]].The trigonometric functions of can be given explicitly as the polynomial roots(4)(5)(6)(7)(8)(9)From one of the Newton-Girard formulas,(10)(11)(12)The trigonometric functions of also obey the identity(13)..

Angle addition formulas express trigonometric functions of sums of angles in terms of functions of and . The fundamental formulas of angle addition in trigonometry are given by(1)(2)(3)(4)(5)(6)The first four of these are known as the prosthaphaeresisformulas, or sometimes as Simpson's formulas.The sine and cosine angle addition identities can be compactly summarized by thematrix equation(7)These formulas can be simply derived using complex exponentials and the Euler formula as follows.(8)(9)(10)(11)Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting for .Taking the ratio of (1) and (3) gives the tangent angle addition formula(12)(13)(14)(15)The double-angle formulas are(16)(17)(18)(19)(20)Multiple-angle formulas are given by(21)(22)and can also be written using the recurrence relations(23)(24)(25)The angle addition formulas can also be derived purely algebraically..

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)To derive these formulas, use the half-angle formulas(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)

The Wallis formula follows from the infinite productrepresentation of the sine(1)Taking gives(2)so(3)(4)(OEIS A052928 and A063196).An accelerated product is given by(5)(6)where(7)(Guillera and Sondow 2005, Sondow 2005). This is analogous to the products(8)and(9)(Sondow 2005).A derivation of equation (◇) due to Y. L. Yung (pers. comm., 1996; modified by J. Sondow, pers. comm., 2002) defines(10)(11)(12)where is a polylogarithm and is the Riemann zeta function, which converges for . Taking the derivative of (11) gives(13)which also converges for , and plugging in then gives(14)(15)(16)Now, taking the derivative of the zeta function expression (◇) gives(17)and again setting yields(18)(19)(20)(21)where(22)(OEIS A075700) follows from the Hadamard product for the Riemann zeta function. Equating and squaring (◇) and (◇) then gives the Wallis formula.This derivation of the..

Trigonometric functions of radians for an integer not divisible by 3 (e.g., and ) cannot be expressed in terms of sums, products, and finite root extractions on rational numbers because 9 is not a product of distinct Fermat primes. This also means that the regular nonagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived using the trigonometric identity(1)Let and . Then the above identity gives the cubic equation(2)(3)This cubic is of the form(4)where(5)(6)The polynomial discriminant is then(7)There are therefore three real distinct roots, which are approximately , 0.3420, and 0.6428. We want the one in the first quadrant, which is approximately 0.3420.(8)(9)(10)(11)Similarly,(12)(13)Because of the Vieta's formulas, we have the identities(14)(15)(16)(15) is known as Morrie's law.Ramanujan found the interesting identity(17)(Borwein and Bailey 2003, p. 77;..

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)To derive these formulas, use the half-angle formulas(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)

The Prosthaphaeresis formulas, also known as Simpson's formulas, are trigonometry formulas that convert a product of functions into a sum or difference. They are given by(1)(2)(3)(4)This form of trigonometric functions can be obtained in the WolframLanguage using the command TrigFactor[expr].These can be derived using the above figure (Kung 1996). From the figure, define(5)(6)Then we have the identity(7)(8)(9)(10)Trigonometric product formulas for the difference of the cosines and sines of two angles can be derived using the similar figure illustrated above (Kung 1996). With and as previously defined, the above figure gives(11)(12)(13)(14)

Let a triangle have side lengths , , and with opposite angles , , and . Then(1)(2)(3)

Construction of the angle produces a 30-60-90 triangle, which has angles and . From the above diagram, write for the vertical leg, then the horizontal leg is given by(1)by the Pythagorean theorem. Now use the double-angle formula(2)to obtain(3)which can be solved for to yield(4)Filling in the rest of the trigonometric functions then gives(5)(6)(7)(8)(9)(10)

The trigonometric formulas for can be derived using the multiple-angle formula(1)Letting and then gives(2)Factoring out one power of gives(3)Solving the quadratic equation for gives(4)But must be less than(5)so taking the minus sign and simplifying gives(6)Filling in the remainder of the trigonometry functions then gives(7)(8)(9)(10)(11)(12)and(13)(14)(15)(16)(17)(18)

Trigonometric functions of for prime have an especially complicated Galois-minimal representation. In particular, the case requires approximately 500 MB of space using the Wolfram Language command Developer`TrigToRadicals[Cos[Pi/23]]. However, they can be expressed concisely as algebraic numbers. For example, letting denote the th root of the polynomial using the ordering of the Wolfram Language's Root function, is given byand by

Construction of the angle produces an isosceles right triangle. Since the sides are equal,(1)so solving for immediately gives(2)Filling in the rest of the trigonometric functions then gives(3)(4)(5)(6)(7)(8)

Values of the trigonometric functions can be expressed exactly for integer multiples of . For ,(1)(2)(3)(4)for ,(5)(6)(7)(8)for ,(9)(10)(11)(12)for ,(13)(14)(15)(16)for ,(17)(18)(19)(20)and for ,(21)(22)(23)(24)These can be derived from the half-angle formulas(25)(26)(27)(28)(29)(30)(31)

Construction of the angle produces a 30-60-90 triangle, which has angles and . From the above diagram, write for the vertical leg, then the horizontal leg is given by(1)by the Pythagorean theorem. Now use the double-angle formula(2)to obtain(3)which can be solved for to yield(4)Filling in the remainder of the trigonometric functions then gives(5)(6)(7)(8)(9)(10)

It is always possible to write a sum of sinusoidal functions(1)as a single sinusoid the form(2)This can be done by expanding (2) using the trigonometricaddition formulas to obtain(3)Now equate the coefficients of (1)and (3)(4)(5)so(6)(7)and(8)(9)giving(10)(11)Therefore,(12)(Nahin 1995, p. 346).In fact, given two general sinusoidal functions with frequency ,(13)(14)their sum can be expressed as a sinusoidal function with frequency (15)(16)(17)Now, define(18)(19)Then (17) becomes(20)Square and add (◇) and (◇)(21)Also, divide (◇) by (◇)(22)so(23)where and are defined by (◇) and (◇).This procedure can be generalized to a sum of harmonic waves, giving(24)(25)where(26)(27)and(28)

The exact values of and can be given by infinite nested radicalswhere the sequence of signs +, +, repeats with period 3, andwhere the sequence of signs , , + repeats with period 3.

By the definition of the functions of trigonometry, the sine of is equal to the -coordinate of the point with polar coordinates , giving . Similarly, , since it is the -coordinate of this point. Filling out the other trigonometric functions then gives(1)(2)(3)(4)(5)(6)

Rather surprisingly, trigonometric functions of for an integer can be expressed in terms of sums, products, and finite root extractions because 17 is a Fermat prime. This makes the heptadecagon a constructible, as first proved by Gauss. Although Gauss did not actually explicitly provide a construction, he did derive the trigonometric formulas below using a series of intermediate variables from which the final expressions were then built up.Let(1)(2)(3)(4)(5)then(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)There are some interesting analytic formulas involving the trigonometric functions of . Define(20)(21)(22)(23)(24)where or 4. Then(25)(26)Another interesting identity is given by(27)where both sides are equal to(28)(Wickner 1999).

By the definition of the functions of trigonometry, the sine of is equal to the -coordinate of the point with polar coordinates , giving . Similarly, , since it is the -coordinate of this point. Filling out the other trigonometric functions then gives(1)(2)(3)(4)(5)(6)

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)These can be derived from the half-angle formulas(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)

Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 7 is not a Fermat prime. This also means that the heptagon is not a constructible polygon.However, exact expressions involving roots of complex numbers can still bederived either using the trigonometric identity(1)with or by expressing in terms of complex exponentials and simplifying the resulting expression. Letting denote the th root of the polynomial using the ordering of the Wolfram Language's Root function gives the following algebraic root representations for trigonometric functions with argument ,(2)(3)(4)(5)(6)(7)with argument ,(8)(9)(10)(11)(12)(13)and with argument ,(14)(15)(16)(17)(18)(19)Root and Galois-minimal expressions can be obtained using WolframLanguage code such as the following: RootReduce[TrigToRadicals[Sin[Pi/7]]] Developer`TrigToRadicals[Sin[Pi/7]]Combinations..

(1)(2)(3)where is a gamma function and is a double factorial.

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