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Trisectible angles | divisibility

  • Thread starter kingwinner
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1) We know that if [itex]\theta[/itex] is trisectible (with straightedge and compass), then [itex]\theta[/itex]/3 is constructible.

But is it also true that if [itex]\theta[/itex]/3 is constructible, then [itex]\theta[/itex] is trisectible (with straightedge and compass)?

If so, then I can say that since 15o is constructible, we have that 45 o is trisectible, right? (because we can copy an angle of 15o three times, thus trisecting the angle 45 o)


2) Let m,n be integers.
Then m|3n3 => m|n
and n|28n3 => n|m

I spent half an hour thinking about this, but I still have no clue...
Why are the implications (=>) true? Can someone please explain?


3) How can I prove that the acute angle whose cosine is 1/10 is constructible?
I know that if [itex]\theta[/itex] is constructible, then cos[itex]\theta[/itex] is constructible. But is the converse true? Why or why not?

Any help is appreciated!:smile:
 
Last edited:

Answers and Replies

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For 1) your logic seems right. For 2), I don't see how the statements are true--are there any other qualifying statements? For example, in the first case, what if m = n2? Or what if m = 3 and n = 2?
 
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1) So is it true that [itex]\theta[/itex] is trisectible (with straightedge and compass) IF AND ONLY IF [itex]\theta[/itex]/3 is constructible (with straightedge and compass)?

2) The whole situtation is this:
http://www.geocities.com/asdfasdf23135/absmath1.jpg
I circled the parts in red which corresopnds to what I've included in my top post.
I don't understand why:
m|3n3 => m|n
and n|28n3 => n|m
where m,n are integers.



Can anyone help?
 
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Can someone please help me with Q3 as well?

I am sure someone here understands it. Please help!!
 
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Still wondering...
 
206
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Isn't your first question essentially, "Can you construct an integer multiple of a constructable angle?" Well...can you?
 
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1) I think that if [itex]\theta[/itex]/3 is constructible, then we can trisect [itex]\theta[/itex] with straightedge and compass by copying the angle [itex]\theta[/itex]/3 two times (since we can always copy any angle with straightedge and compass)
 
tiny-tim
Science Advisor
Homework Helper
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… cos^-1 of constructible number is constructible angle …

3) How can I prove that the acute angle whose cosine is 1/10 is constructible?
Hi kingwinner! :smile:

Any angle whose cosine is a constructible number between -1 and 1 (like 4/5 or 1/√2) is constructible!

Hint: draw a circle. Draw one radius. Mark 1/10 along that radius. And then … ? :smile:
 
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Hi kingwinner! :smile:

Any angle whose cosine is a constructible number between -1 and 1 (like 4/5 or 1/√2) is constructible!

Hint: draw a circle. Draw one radius. Mark 1/10 along that radius. And then … ? :smile:
And then erect a pernpendicular at that point to consturuct the angle??? (since on the unit circle, x=cos(theta), where theta is counterclockwise from positive x-axis)
 
tiny-tim
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Homework Helper
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Yes!!!! :smile:

(… why only three question marks? …)
 
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3) So we have theta constructible if and only if cos(theta) is constructible


2) Let m,n be integers
m|3n3 => m|n
and n|28n3 => n|m
Do you think these are actually wrong implications? (i.e. whoever was writing the solutions got it wrong...)
 
206
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Without additional assumptions on m and n, the implications aren't true...
 

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