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  2. Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Euler's_formula

    Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives ⁠ dr / dx ⁠ = 0 and ⁠ dθ / dx ⁠ = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x.

  3. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, (⁡ + ⁡) = ⁡ + ⁡. If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]

  4. Gaussian integral - Wikipedia

    en.wikipedia.org/wiki/Gaussian_integral

    A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that ex 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.

  5. Integration using Euler's formula - Wikipedia

    en.wikipedia.org/wiki/Integration_using_Euler's...

    At this point we can either integrate directly, or we can first change the integrand to 2 cos 6x − 4 cos 4x + 2 cos 2x and continue from there. Either method gives Either method gives ∫ sin 2x cos ⁡ 4 x d x = − 1 24 sin ⁡ 6 x + 1 8 sin ⁡ 4 x − 1 8 sin ⁡ 2 x + C . {\displaystyle \int \sin ^{2}x\cos 4x\,dx=-{\frac {1}{24 ...

  6. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 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.

  7. cis (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Cis_(mathematics)

    x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, e ix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.

  8. List of integrals of trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/List_of_integrals_of...

    The following is a list of integrals (antiderivative functions) of trigonometric functions.For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions.

  9. Chebyshev polynomials - Wikipedia

    en.wikipedia.org/wiki/Chebyshev_polynomials

    The real part of the other side is a polynomial in cos x and sin x, in which all powers of sin x are even and thus replaceable through the identity cos 2 x + sin 2 x = 1. By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one factor of sin x is factored out, the remaining ...