Write the inner product for the case m =n and integrate. 2n | sin mt cos mt dt = Evaluate the integral. 2n sin mt cos mt dt = (Simplify your answer.) Therefore, sin mt and cos nt are orthogonal for...

Extracted text: Write the inner product for the case m =n and integrate. 2n | sin mt cos mt dt = Evaluate the integral. 2n sin mt cos mt dt = (Simplify your answer.) Therefore, sin mt and cos nt are orthogonal for all positive integers m and n because the inner product is always


Extracted text: 2n The space is C[0,2r] and the inner product is (f.g) = f(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n. Begin by writing the inner product using the given functions. (sin mt, cos nt) = dt Use a trigonometric identity to write the integrand as a sum of sines. 2x 1 (sin mt, cos nt) =- dt Then integrate with respect to t. (sin mt, cos nt) = Evaluate the result at the end points of the interval. Note that m -n in the denominator means that this result does not apply to m = n. (sin mt, cos nt) = [(O - 0) (Simplify your answers.) Then simplify this result to get the inner product for all positive integers m # n. (sin mt, cos nt) =