I'm trying to find all intervals [ a , b ] on which the functions sin &#x20

anginih86 2022-06-24 Answered
I'm trying to find all intervals [ a , b ] on which the functions sin ( 2 π t ) and cos ( 2 p i t ) are orthogonal.
a b sin ( 2 π t ) cos ( 2 π t ) d t = cos ( 4 π b ) c o s ( 4 π a ) 8 π = 0
cos ( 4 π b + 2 π k ) = c o s ( 4 π a + 2 π l ) , k , l Z
I don't know how to solve this for a and b, can anybody help me with that please?
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Answers (1)

Tianna Deleon
Answered 2022-06-25 Author has 29 answers
cos ( 4 π a ) cos ( 4 π b ) 8 π = 0
cos ( 4 π a ) cos ( 4 π b ) = 0
cos ( 4 π b ) = cos ( 4 π a )
cos ( 4 π b ) = cos ( 4 π a )
4 π b = 4 π a + 2 π n 1         4 π b = 2 π n 1 4 π a
b = a + n 1 2         b = n 2 2 a
With n 1 , n 2 Z
So you can set:
b = ± a + n 2   , n Z

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