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For example, the derivative of the sine function is written sin ′ (a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as ...
The derivative of sine is cosine, and the derivative of cosine is negative sine: [16] = (), = (). Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. [ 15 ]
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.
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 derivative of order zero of f is defined to be f itself and ... The polynomials, exponential function e x, and the trigonometric functions sine and cosine, ...
For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin θ < θ. So we have < <. For negative values of θ we have, by the symmetry of the sine function
The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(ξ) / ξ = cos(ξ) for all points ξ where the derivative of sin(x) / x is zero and thus a local extremum is reached.
A similar proof can be completed using power series as above to establish that the sine has as its derivative the cosine, and the cosine has as its derivative the negative sine. In fact, the definitions by ordinary differential equation and by power series lead to similar derivations of most identities.