In this paper we study two families of functions, viz. F and H, and show how to approximate the functions considered in the interval [0,1 ]. The functions are assumed to be real when the argument is real. We define \[ F = { f ; ( i ) f ( 1 2 + x ) = f ( 1 2 − x ) , ( ii ) f ( 0 ) = f ( 1 ) = 0 , ( iii ) f ( x ) is analytic in a sufficiently large neighborhood of x = 0 } , F = \{ f;({\text {i}})\,f\left ( {\frac {1}{2} + x} \right ) = f\left ( {\frac {1}{2} - x} \right ),({\text {ii}})\,f(0) = f(1) = 0,({\text {iii}})\;f(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}, \] \[ H = { h ; ( j ) h ( 1 2 + x ) = − h ( 1 2 − x ) , ( jj ) h ( 0 ) = h ( 1 ) = 0 , ( jjj ) h ( x ) is analytic in a sufficiently large neighborhood of x = 0 } . H = \{ h;({\text {j}})\;h\left ( {\frac {1}{2} + x} \right ) = - h\left ( {\frac {1}{2} - x} \right ),({\text {jj}})\;h(0) = h(1) = 0,({\text {jjj}})\;h(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}. \] The approximations are defined in the interval [0,1 ] by \[ min ∫ 0 1 ( f ( x ) − ∑ n = 1 k c n , k [ x ( 1 − x ) ] n ) 2 x q ( 1 − x ) q d x \min \int _0^1 {{{\left ( {f(x) - \sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} \] and \[ min ∫ 0 1 ( h ( x ) − ( 1 − 2 x ) ∑ n = 1 k c n , k [ x ( 1 − x ) ] n ) 2 x q ( 1 − x ) q d x , \min \int _0^1 {{{\left ( {h(x) - (1 - 2x)\sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} , \] where q ∈ { 0 , 1 , 2 , … } q \in \{ 0,1,2, \ldots \} . The associated matrices are analyzed using the theory of orthogonal polynomials, especially the Jacobi polynomials G n ( p , q , x ) {G_n}(p,q,x) . We apply the general theory to the basic trigonometric functions sin ( x ) \sin (x) and cos ( x ) \cos (x) .