Sine and cosine
Series and polynomials

Both sine and cosine functions can be defined by using a Taylor series, a power series involving the higher-order derivatives. As mentioned in § Continuity and differentiation, the derivative of sine is cosine and that the derivative of cosine is the negative of sine. This means the successive derivatives of sin (x) {\displaystyle \sin(x)} are cos (x) {\displaystyle \cos(x)}, − sin (x) {\displaystyle -\sin(x)}, − cos (x) {\displaystyle -\cos(x)}, sin (x) {\displaystyle \sin(x)}, continuing to repeat those four functions. The (4 n + k) {\displaystyle (4n+k)} - th derivative, evaluated at the point 0: sin (4 n + k) (0) = { 0 when k = 0 1 when k = 1 0 when k = 2 − 1 when k = 3 {\displaystyle \sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}} where the superscript represents repeated differentiation. This implies the following Taylor series expansion at x = 0 {\displaystyle x=0}. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x {\displaystyle x} — where x {\displaystyle x} is the angle in radians. More generally, for all complex numbers: sin (x) = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = ∑ n = 0 ∞ (− 1) n (2 n + 1) ! x 2 n + 1 {\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}} Taking the derivative of each term gives the Taylor series for cosine: cos (x) = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ = ∑ n = 0 ∞ (− 1) n (2 n) ! x 2 n {\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}}

Both sine and cosine functions with multiple angles may appear as their linear combination, resulting in a polynomial. Such a polynomial is known as the trigonometric polynomial. The trigonometric polynomial's ample applications may be acquired in its interpolation, and its extension of a periodic function known as the Fourier series. Let a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} be any coefficients, then the trigonometric polynomial of a degree N {\displaystyle N} — denoted as T (x) {\displaystyle T(x)} — is defined as: T (x) = a 0 + ∑ n = 1 N a n cos (n x) + ∑ n = 1 N b n sin (n x). {\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n=1}^{N}b_{n}\sin(nx).}

The trigonometric series can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let A n {\displaystyle A_{n}} and B n {\displaystyle B_{n}} be any coefficients, then the trigonometric series can be defined as: 1 2 A 0 + ∑ n = 1 ∞ A n cos (n x) + B n sin (n x). {\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).} In the case of a Fourier series with a given integrable function f {\displaystyle f}, the coefficients of a trigonometric series are: A n = 1 π ∫ 0 2 π f (x) cos (n x) d x, B n = 1 π ∫ 0 2 π f (x) sin (n x) d x. {\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\sin(nx)\,dx.\end{aligned}}}
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