Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Trigonometric functions of matrices
open-in-new
<p>The trigonometric functions (especially <a href="/facts/Sine/gQ6LXK8z">sine</a> and <a href="/facts/Cosine/czej7ur0">cosine</a>) for complex <a href="/facts/Square_matrices/5TP9mNNn">square matrices</a> occur in solutions of second-order systems of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>. They are defined by the same <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> that hold for the trigonometric functions of <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>: </p> sin X = X − X 3 3 ! + X 5 5 ! − X 7 7 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! X 2 n + 1 cos X = I − X 2 2 ! + X 4 4 ! − X 6 6 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! X 2 n {\displaystyle {\begin{aligned}\sin X&=X-{\frac {X^{3}}{3!}}+{\frac {X^{5}}{5!}}-{\frac {X^{7}}{7!}}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}X^{2n+1}\\\cos X&=I-{\frac {X^{2}}{2!}}+{\frac {X^{4}}{4!}}-{\frac {X^{6}}{6!}}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}X^{2n}\end{aligned}}} <p>with <i>Xn</i> being the nth <a href="/facts/Matrix_multiplication/lYDB6Ro2">power</a> of the matrix X, and I being the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> of appropriate dimensions. </p><p>Equivalently, they can be defined using the <a href="/facts/Matrix_exponential/WID5cG2g">matrix exponential</a> along with the matrix equivalent of <a href="/facts/Euler%27s_formula/Wq7VXIV0">Euler's formula</a>, <i>eiX</i> = cos <i>X</i> + <i>i</i> sin <i>X</i>, yielding </p> sin X = e i X − e − i X 2 i cos X = e i X + e − i X 2 . {\displaystyle {\begin{aligned}\sin X&={e^{iX}-e^{-iX} \over 2i}\\\cos X&={e^{iX}+e^{-iX} \over 2}.\end{aligned}}} <p>For example, taking X to be a standard <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrix</a>, </p> σ 1 = σ x = ( 0 1 1 0 ) , {\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}~,} <p>one has </p> sin ( θ σ 1 ) = sin ( θ ) σ 1 , cos ( θ σ 1 ) = cos ( θ ) I , {\displaystyle \sin(\theta \sigma _{1})=\sin(\theta )~\sigma _{1},\qquad \cos(\theta \sigma _{1})=\cos(\theta )~I~,} <p>as well as, for the <a href="/facts/Sinc_function/lSvDtHy0">cardinal sine function</a>, </p> sinc ( θ σ 1 ) = sinc ( θ ) I . {\displaystyle \operatorname {sinc} (\theta \sigma _{1})=\operatorname {sinc} (\theta )~I.}