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Let P = (x, y) and let ψ be the angle between SB and the x-axis; this is equal to the angle between ST and J. By construction, PT = a, so the distance from P to J is a sin ψ. In other words a – x = a sin ψ. Also, SP = a is the y-coordinate of (x, y) if it is rotated by angle ψ, so a = (x + a) sin ψ + y cos ψ. After simplification, this ...
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.
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to: = + . Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
ψ (psi) is the angle in the chain N − C α − C' − N (called φ′ by Ramachandran) The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined). [8] The planarity of the peptide bond usually restricts ω to be 180° (the typical trans case) or 0° (the rare cis case).
satisfying respectively y(0) = 0, y ′ (0) = 1 and y(0) = 1, y ′ (0) = 0. It follows from the theory of ordinary differential equations that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that ...
Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9). Numerical solution of the KdV equation u t + uu x + δ 2 u xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx).
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
The failure of classical mechanics applied to molecular, atomic, and nuclear systems and smaller induced the need for a new mechanics: quantum mechanics.The mathematical formulation was led by De Broglie, Bohr, Schrödinger, Pauli, and Heisenberg, and others, around the mid-1920s, and at that time was analogous to that of classical mechanics.