Formula for $cos((2n+1)x)$ as polynomial of $cos x$
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I am looking for a formula of $cos((2n+1)x)$ that is polynomial of $cos(x)$.
For example, $$cos3alpha=4cos^3alpha-3cosalpha$$ Is it known for any $n$?
trigonometry
edited Nov 15 at 6:34
Robert Howard
1,553622
asked Oct 12 '14 at 19:31
Aldor
62
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show 1 more comment
up vote
1
down vote
favorite
I am looking for a formula of $cos((2n+1)x)$ that is polynomial of $cos(x)$.
For example, $$cos3alpha=4cos^3alpha-3cosalpha$$ Is it known for any $n$?
trigonometry
edited Nov 15 at 6:34
Robert Howard
1,553622
asked Oct 12 '14 at 19:31
Aldor
62
1
en.wikipedia.org/wiki/Chebyshev_polynomials
– Hans Lundmark
Oct 12 '14 at 19:33
@HansLundmark Beat me to it ...
– Mark Bennet
Oct 12 '14 at 19:34
@HansLundmark great, thanks a lot
– Aldor
Oct 12 '14 at 19:54
@HansLundmark Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue?
– Robert Howard
Nov 15 at 5:02
1
@RobertHoward: OK, done.
– Hans Lundmark
Nov 15 at 5:37
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am looking for a formula of $cos((2n+1)x)$ that is polynomial of $cos(x)$.
For example, $$cos3alpha=4cos^3alpha-3cosalpha$$ Is it known for any $n$?
trigonometry
edited Nov 15 at 6:34
Robert Howard
1,553622
asked Oct 12 '14 at 19:31
Aldor
62
I am looking for a formula of $cos((2n+1)x)$ that is polynomial of $cos(x)$.
For example, $$cos3alpha=4cos^3alpha-3cosalpha$$ Is it known for any $n$?
trigonometry
trigonometry
edited Nov 15 at 6:34
Robert Howard
1,553622
asked Oct 12 '14 at 19:31
Aldor
62
edited Nov 15 at 6:34
Robert Howard
1,553622
asked Oct 12 '14 at 19:31
Aldor
62
edited Nov 15 at 6:34
Robert Howard
1,553622
edited Nov 15 at 6:34
Robert Howard
1,553622
edited Nov 15 at 6:34
Robert Howard
1,553622
1,553622
asked Oct 12 '14 at 19:31
Aldor
62
asked Oct 12 '14 at 19:31
Aldor
62
asked Oct 12 '14 at 19:31
Aldor
62
62
1
en.wikipedia.org/wiki/Chebyshev_polynomials
– Hans Lundmark
Oct 12 '14 at 19:33
@HansLundmark Beat me to it ...
– Mark Bennet
Oct 12 '14 at 19:34
@HansLundmark great, thanks a lot
– Aldor
Oct 12 '14 at 19:54
@HansLundmark Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue?
– Robert Howard
Nov 15 at 5:02
1
@RobertHoward: OK, done.
– Hans Lundmark
Nov 15 at 5:37
|
show 1 more comment
1
en.wikipedia.org/wiki/Chebyshev_polynomials
– Hans Lundmark
Oct 12 '14 at 19:33
@HansLundmark Beat me to it ...
– Mark Bennet
Oct 12 '14 at 19:34
@HansLundmark great, thanks a lot
– Aldor
Oct 12 '14 at 19:54
@HansLundmark Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue?
– Robert Howard
Nov 15 at 5:02
1
@RobertHoward: OK, done.
– Hans Lundmark
Nov 15 at 5:37
1
1
en.wikipedia.org/wiki/Chebyshev_polynomials
– Hans Lundmark
Oct 12 '14 at 19:33
en.wikipedia.org/wiki/Chebyshev_polynomials
– Hans Lundmark
Oct 12 '14 at 19:33
@HansLundmark Beat me to it ...
– Mark Bennet
Oct 12 '14 at 19:34
@HansLundmark Beat me to it ...
– Mark Bennet
Oct 12 '14 at 19:34
@HansLundmark great, thanks a lot
– Aldor
Oct 12 '14 at 19:54
@HansLundmark great, thanks a lot
– Aldor
Oct 12 '14 at 19:54
@HansLundmark Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue?
– Robert Howard
Nov 15 at 5:02
@HansLundmark Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue?
– Robert Howard
Nov 15 at 5:02
1
1
@RobertHoward: OK, done.
– Hans Lundmark
Nov 15 at 5:37
@RobertHoward: OK, done.
– Hans Lundmark
Nov 15 at 5:37
|
show 1 more comment
1 Answer
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These are the Chebyshev polynomials of the first kind, usually denoted $T_n(x)$.
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
These are the Chebyshev polynomials of the first kind, usually denoted $T_n(x)$.
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
add a comment |
up vote
1
down vote
These are the Chebyshev polynomials of the first kind, usually denoted $T_n(x)$.
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
add a comment |
up vote
1
down vote
up vote
1
down vote
These are the Chebyshev polynomials of the first kind, usually denoted $T_n(x)$.
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
These are the Chebyshev polynomials of the first kind, usually denoted $T_n(x)$.
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
answered Nov 15 at 5:36
Hans Lundmark
34.4k564110
34.4k564110
add a comment |
add a comment |
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en.wikipedia.org/wiki/Chebyshev_polynomials
– Hans Lundmark
Oct 12 '14 at 19:33
@HansLundmark Beat me to it ...
– Mark Bennet
Oct 12 '14 at 19:34
@HansLundmark great, thanks a lot
– Aldor
Oct 12 '14 at 19:54
@HansLundmark Could you please convert your comment into an answer so this question can be removed from the "Unanswered" queue?
– Robert Howard
Nov 15 at 5:02
1
@RobertHoward: OK, done.
– Hans Lundmark
Nov 15 at 5:37