Basic version of Fourier polynomials.(a) Write down the trig identities forcos(x + y) = ..., cos(x – y) = ..., sin(r+y) =..., sin(r – y) = ..(b) Using part (a) derive (explain all the steps) the trig identitiescos x cos y = ..., sinx sin y = ..., sin x cos y =...(c) Assume that m,n are nonnegative integers (zero included!). Using part (b) evaluate thefollowing integrals:sin mæ sin nædr, | acos 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 finitelinear 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.

Answered: Basic version of Fourier polynomials.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

100%

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.

Expert Solution

Step 1

Since, you have asked question with many sub parts, so, I will be answering the first three sub-parts. please re-post the last part separately.

(a)

The trigonometric identities are as follows:

$\cos (x + y) = \cos (x) \cdot \cos (y) - \sin (x) \cdot \sin (y); \dots \dots \dots (i) \cos (x - y) = \cos (x) \cdot \cos (y) + \sin (x) \cdot \sin (y); \dots \dots \dots (i i) \sin (x + y) = \sin (x) \cdot \cos (y) + \sin (y) \cdot \cos (x); \dots \dots \dots (i i i) \sin (x - y) = \sin (x) \cdot \cos (y) - \sin (y) \cdot \cos (x) \dots \dots \dots (i v)$

(b)

Adding (i) and (ii), we get

$\cos (x + y) + \cos (x - y) = 2 \cdot \cos (x) \cdot \cos (y) \Rightarrow \cos (x) \cdot \cos (y) = \frac{\cos (x + y) + \cos (x - y)}{2} \dots \dots \dots \dots (v)$

Subtract (i) from (ii), we get

$\cos (x - y) - \cos (x + y) = 2 \cdot \sin (x) \cdot \sin (y) \Rightarrow \sin (x) \cdot \sin (y) = \frac{\cos (x - y) - \cos (x + y)}{2} \dots \dots \dots \dots (v i)$

Adding (iii) and (iv), we get

$\sin (x + y) + \sin (x - y) = 2 \cdot \sin (x) \cdot \cos (y) \Rightarrow \sin (x) \cdot \cos (y) = \frac{\sin (x + y) + \sin (x - y)}{2} \dots \dots \dots \dots (v i i)$

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Fourier Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions