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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.
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.
In this right triangle, denoting the measure of angle BAC as A: sin A = a / c ; cos A = b / c ; tan A = a / b . Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labeled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point.
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...
In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves ), and in signal processing (to describe FM signals).
This sum is called a Chebyshev series or a Chebyshev expansion. Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart. [ 16 ]
The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A.As is shown, H and A are almost the same length, meaning cos θ is close to 1 and θ 2 / 2 helps trim the red away.
Note that h 2 is the power of the point B with respect to the circle. The use of the Pythagorean theorem and the tangent secant theorem can be replaced by a single application of the power of a point theorem. Case of acute angle γ, where a < 2b cos γ. Drop the perpendicular from A onto a = BC, creating a line segment of length b cos γ.