# Trisectible angles | divisibility

1. Mar 18, 2008

### 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 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

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 $\theta$ is constructible, then cos$\theta$ is constructible. But is the converse true? Why or why not?

Any help is appreciated!

Last edited: Mar 18, 2008
2. Mar 18, 2008

### 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?

3. Mar 20, 2008

### 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?

4. Mar 22, 2008

### kingwinner

5. Mar 24, 2008

### kingwinner

Still wondering...

6. Mar 24, 2008

### Mystic998

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

7. Mar 25, 2008

### 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)

8. Mar 25, 2008

### tiny-tim

… cos^-1 of constructible number is constructible angle …

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 … ?

9. Mar 25, 2008

### kingwinner

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)

10. Mar 25, 2008

### tiny-tim

Yes!!!!

(… why only three question marks? …)

11. Mar 25, 2008

### 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...)

12. Mar 25, 2008

### Mystic998

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