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Let us assume that given function f can be represented on the interval [-T,7] as a finite linear combination of constant, sines, and cosines: f(r) = ao + (ak cos kz + bg sin kr). k=1 Using part (c) find the expressions for ao, ak, br in terms of f(x). Hint: Multiply both sides of the equality with an appropriate function and integrate from -a to n. A lot of terms should disappear.

Let us assume that given function f can be represented on the interval [-T,7] as a finite linear combination of constant, sines, and cosines: f(r) = ao + (ak cos kz + bg sin kr). k=1 Using part (c) find the expressions for ao, ak, br in terms of f(x). Hint: Multiply both sides of the equality with an appropriate function and integrate from -a to n. A lot of terms should disappear.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 70E
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PART D ONLY PLEASE

Basic version of Fourier polynomials. (a) Write down the trig identities for cos(x + y) = ..., cos(x – y) = ..., sin(r+y) =..., sin(r – y) = .. (b) Using part (a) derive (explain all the steps) the trig identities cos x cos y = ..., sinx sin y = ..., sin x cos y =... (c) Assume that m,n are nonnegative integers (zero included!). Using part (b) evaluate the following integrals: sin mæ sin nædr, | a cos mr cos nad, sin mæ cos nadr. Hint: You should be careful and consider different cases: 1) n = 0, m = 0, 2) n = 0, m > 0, 3) n > 0, m = 0, 4) n > 0, m > 0, m = n, and 5) n > 0, m > 0, m ± n. (d) Let us assume that given function f can be represented on the interval [-7, 7] as a finite linear combination of constant, sines, and cosines: f(x) = ao + (ar cos ka + b̟ sin kæ). Using part (c) find the expressions for ao, ak, bk in terms of f(x). Hint: Multiply both sides of the equality with an appropriate function and integrate from -n to T. A lot of terms should disappear.
Transcribed Image Text:Basic version of Fourier polynomials. (a) Write down the trig identities for cos(x + y) = ..., cos(x – y) = ..., sin(r+y) =..., sin(r – y) = .. (b) Using part (a) derive (explain all the steps) the trig identities cos x cos y = ..., sinx sin y = ..., sin x cos y =... (c) Assume that m,n are nonnegative integers (zero included!). Using part (b) evaluate the following integrals: sin mæ sin nædr, | a cos mr cos nad, sin mæ cos nadr. Hint: You should be careful and consider different cases: 1) n = 0, m = 0, 2) n = 0, m > 0, 3) n > 0, m = 0, 4) n > 0, m > 0, m = n, and 5) n > 0, m > 0, m ± n. (d) Let us assume that given function f can be represented on the interval [-7, 7] as a finite linear combination of constant, sines, and cosines: f(x) = ao + (ar cos ka + b̟ sin kæ). Using part (c) find the expressions for ao, ak, bk in terms of f(x). Hint: Multiply both sides of the equality with an appropriate function and integrate from -n to T. A lot of terms should disappear.
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