Trisectible angles | divisibility

[kingwinner]

1) We know that if \theta is trisectible (with straightedge and compass), then \theta/3 is constructible.

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

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


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

I spent half an hour thinking about this, but I still have no clue...
Why are the implications (=>[/color]) 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 \theta is constructible, then cos\theta is constructible. But is the converse true? Why or why not?

Any help is appreciated!

 

[Tedjn]

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?

 

[kingwinner]

1) So is it true that \theta is trisectible (with straightedge and compass) IF AND ONLY IF \theta/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?

 

[kingwinner]

Can someone please help me with Q3 as well?

I am sure someone here understands it. Please help!

 

[kingwinner]

Still wondering...

 

[Mystic998]

Isn't your first question essentially, "Can you construct an integer multiple of a constructable angle?" Well...can you?

 

[kingwinner]

1) I think that if \theta/3 is constructible, then we can trisect \theta with straightedge and compass by copying the angle \theta/3 two times (since we can always copy any angle with straightedge and compass)

 

[tiny-tim]

… 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!

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 … ?

 

[kingwinner]

Hi kingwinner!

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 … ?

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]

Yes!

(… why only three question marks? …)

 

[kingwinner]

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...)

 

[Mystic998]

Without additional assumptions on m and n, the implications aren't true...