Algebraic cosine

[Genady]

 

\(\cos(\frac {\pi} {4})=\frac {1} {\sqrt{2}}\)

\(\cos(\frac {\pi} {5})=\frac {1+\sqrt{5}} {4}\)

\(\cos(\frac {\pi} {6})=\frac {\sqrt{3}} {2}\)

Is cosine of any fraction of π an algebraic number?

Edited December 13 by Genady

[studiot]

Waht is wrong with   pi/3  or 0pi ?

 

But what do you think an algebraic number is ?

 

square root of 2 is algebraic since it is the root of x² -2 = 0

[Genady]

14 minutes ago, studiot said:

Waht is wrong with   pi/3  or 0pi ?

 

But what do you think an algebraic number is ?

 

square root of 2 is algebraic since it is the root of x² -2 = 0

Nothing is wrong with them.

I think it is as defined in textbooks.

Yes, square root of 2 is algebraic.

So, the question remains, is cosine of any fraction of pi algebraic?

The other question is, was my English in the OP correct?

Edited December 13 by Genady

[KJW]

1 hour ago, Genady said:

Is cosine of any fraction of π an algebraic number?

The cosine of any fraction of π is the real part of a root of unity and is called a trigonometric number.

 

Edited December 13 by KJW

[studiot]

Just now, Genady said:

The other question is, was my English in the OP correct?

Well

Just now, Genady said:

Is cosine of any fraction of π an algebraic number?

I would have said, "is every fraction of π ..."

Which is proved in the  excellent Wiki article KJW (+1) linked to.

I took any to mean  "are there any fractions of π ... ?"

Technically any would do but is ambiguous, whereas every is not.

😀

[Genady]

1 hour ago, KJW said:

 trigonometric number.

Thank you. It answers the question.

 

37 minutes ago, studiot said:

Technically any would do but is ambiguous, whereas every is not.

Thank you. I will keep it in mind.

[Genady]

On 12/13/2024 at 12:59 PM, studiot said:

any would do but is ambiguous

This reminded me of another ambiguous term, ever. I remember (vaguely) an argument in a class I took many years ago, about a quiz question that was something like, Is the star A ever visible at the location B? Some students answered Yes because it is sometimes visible, while others answered No because it is sometimes invisible. 

[studiot]

Just now, Genady said:

This reminded me of another ambiguous term, ever. I remember (vaguely) an argument in a class I took many years ago, about a quiz question that was something like, Is the star A ever visible at the location B? Some students answered Yes because it is sometimes visible, while others answered No because it is sometimes invisible. 

This is why English has constructions where you need more than one word to exactly specify a meaning.

To me is it ever visible has meaning A alone , to be used in response to a statemnent like

Venus is not visible tonight (because of cloud cover)

Someone might ask

Is it ever visible ?

If the speaker wanted to say the star is visible at all times she would never say the star is 'ever visible' in normal parlance.

(unless she was using poetic licence as in 'the Star of Bethlehem was ever visible to the three wise men')

She might just say the star is/was always visible.

If she was as optimistic as Oztmandias she might say 'the star will be visible for ever'.

When the statements are analysed the tense of the verb must match the rest of the statement, something Russians probably find more difficult as English has many tenses.

 

Other phrases, this time from legal English, demonstrate my point about additional words

The phrase Jointly and Severally to indicate the distribution of responsibility for something.

Similarly we have 'Tenants in Common'  v  'Joint Tenants'  - a very important distinction.

 

[Genady]

11 minutes ago, studiot said:

This is why English has constructions where you need more than one word to exactly specify a meaning.

To me is it ever visible has meaning A alone , to be used in response to a statemnent like

Venus is not visible tonight (because of cloud cover)

Someone might ask

Is it ever visible ?

If the speaker wanted to say the star is visible at all times she would never say the star is 'ever visible' in normal parlance.

(unless she was using poetic licence as in 'the Star of Bethlehem was ever visible to the three wise men')

She might just say the star is/was always visible.

If she was as optimistic as Oztmandias she might say 'the star will be visible for ever'.

When the statements are analysed the tense of the verb must match the rest of the statement, something Russians probably find more difficult as English has many tenses.

 

Other phrases, this time from legal English, demonstrate my point about additional words

The phrase Jointly and Severally to indicate the distribution of responsibility for something.

Similarly we have 'Tenants in Common'  v  'Joint Tenants'  - a very important distinction.

 

Just want to clarify that the class was in Duke University, NC, with American students. And that I was among the ones who understood and answered the question as was intended by the professor, i.e., it is visible sometimes.