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The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers: with Xn being the nth power of the matrix X, and I being the identity matrix of appropriate dimensions. Equivalently, they can be defined using the matrix exponential along with the matrix equivalent of Euler's formula, eiX = cos X + i sin X, yielding For example, taking X to be a standard Pauli matrix, one has

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  • Trigonometric functions of matrices (en)
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  • The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers: with Xn being the nth power of the matrix X, and I being the identity matrix of appropriate dimensions. Equivalently, they can be defined using the matrix exponential along with the matrix equivalent of Euler's formula, eiX = cos X + i sin X, yielding For example, taking X to be a standard Pauli matrix, one has (en)
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  • The trigonometric functions (especially sine and cosine) for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers: with Xn being the nth power of the matrix X, and I being the identity matrix of appropriate dimensions. Equivalently, they can be defined using the matrix exponential along with the matrix equivalent of Euler's formula, eiX = cos X + i sin X, yielding For example, taking X to be a standard Pauli matrix, one has as well as, for the cardinal sine function,(See also: Axis–angle representation § Exponential map from so(3) to SO(3)) (en)
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