Show that the given functions are orthogonal on the indicated interval The image contains mathematical expressions and a specified interval:1. $ f_1(x) = \cos x $2. $ f_2(x) = \sin^2 x $3. The interval specified is $[0, \pi]$.These functions describe typical trigonometric forms within the given interval.

Answered: Show that the given functions are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Show that the given functions are orthogonal on the indicated interval

The image contains mathematical expressions and a specified interval:

1. $ f_1(x) = \cos x $
2. $ f_2(x) = \sin^2 x $
3. The interval specified is $[0, \pi]$.

These functions describe typical trigonometric forms within the given interval.

Expert Solution

Step 1

We have to prove that the functions $f_{1} (x) = \cos x$ and $f_{2} (x) = \sin^{2} x$ are orthogonal on the interval $[0, π]$ .

In order to prove that 2 functions $f_{1} (x)$ and $f_{2} (x)$ are orthogonal on an interval $[a, b]$ , we have to prove that $\int_{a}^{b} f_{1} (x) f_{2} (x) d x = 0$ .

So, to prove that the functions $f_{1} (x) = \cos x$ and $f_{2} (x) = \sin^{2} x$ are orthogonal on the interval $[0, π]$ , we have to prove that $\int_{0}^{π} (\cos x) (\sin^{2} x) d x = 0$ .

First we obtain the integral $\int (\cos x) (\sin^{2} x) d x$ as follows.

Take $t = \sin x$ .

Then, $d t = \cos x d x$ .

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