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This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, ... For these functions the Taylor series do not converge if x is far from b.
The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation , cos θ {\displaystyle \textstyle \cos \theta } is approximated as either 1 {\displaystyle 1} or as 1 − 1 2 θ 2 {\textstyle 1-{\frac {1}{2}}\theta ^{2}} .
The function has also been called the cardinal sine or sine cardinal function. [3] [4] ... The Taylor series of the unnormalized sinc function can be obtained from ...
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.
For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. [1] There are several versions of Taylor's theorem, some ...
The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
The trigonometric functions (especially sine and cosine) for complex square matrices occur in solutions of second-order systems of differential equations. [1] They are defined by the same Taylor series that hold for the trigonometric functions of complex numbers: [2] =! +!! + = = (+)!