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pablos28spainzd

pablos28spainzd

Answered question

2022-07-03

If f ( x ) = cos x cos 2 x cos 4 x cos 8 x cos 16 x, then f ( π / 4 ) = ?

Answer & Explanation

toriannucz

toriannucz

Beginner2022-07-04Added 16 answers

HINT:
Multiply by 2 sin x to get
2 5 sin x f ( x ) = sin 32 x
Observe that f ( π 4 ) = 0
lilmoore11p8

lilmoore11p8

Beginner2022-07-05Added 6 answers

Note that, in general, if f ( x ) = cos x cos 2 x cos 4 x cos 8 x cos 16 x . . . cos 2 n 1 x, then, multiplying by 2 n sin ( x ) and repeatedly using sin ( 2 x ) = 2 sin ( x ) cos ( x ), 2 n sin ( x ) f ( x ) = sin ( 2 n x ).
Note that f ( π / 4 ) = 0 if n 1 since cos ( π / 2 ) = 0
Differentiating, 2 n ( sin ( x ) f ( x ) + cos ( x ) f ( x ) ) = 2 n cos ( 2 n x ), so sin ( x ) f ( x ) + cos ( x ) f ( x ) = cos ( 2 n x )
Setting x = π / 4, and remembering that f ( π / 4 ) = 0, sin ( π / 4 ) f ( π / 4 ) = cos ( 2 n 2 π ) or f ( π / 4 ) = 2 if n 3

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