Question

Product-to-sum and sum-to-product trigonometrie identities Product-to-sum Sum-to-product cos(0-φ) + cos(θ + φ) e-p)-cos sin θ sin so sin(θ +p) + sin(0-4) 2 ) |cos θ sin φ 2/25/2018 Orthogonal set of Sinusoidal Function:s 3.11, cos(nLx)cosenLx)dx={0 we now prove this one n=m#0 πχ sincos dx = 0,V n,m

Proof the following integration using the provided trigonometric identities (please show in clear and neat steps) :

Answer 1

integral^L_- L cos(n pi x/L) dx = [sin(n pi x/L) times L/nn]^L_- L = 0 (Because sin n pi = 0) 2) integral^L_- L sin (n pi x/L) dx = -[L/n pi times cos(n pi x/L)]^L_- L = 0 (Because cos n pi = cos (- n pi) = (- 1)^n) 3) If n notequalto m then, I = 1/2 integral^L_- L 2 cos (n pi x/L) cos(m pi x?L) dx = 1/2 integral^L_- L cos (m + n) pi x/L + cos((n - m) pi x)/L) dx = 1/2[L/(m + n)pi sin(m + n) pi x/L + L/(n - m) pi sin(n - m) pi x/L)]^L_- L = 0 (Because sin(m + n) L pi = sin(n - m) L pi = 0) If m = n then, I = 1/2 integral^L_- L 2 cos^2(n pi x/L) dx = 1/2 integral^L_- L [1 + cos(2 n pi x/L)] dx