Math Is Fun Forum

Kurre

Member

Registered: 2006-07-18

Posts: 280

Integer parts and trigonometry

Solve the following system of equations in real x and y, where angles are measured in radians and [a] means the greatest integer less or equal to a:

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Integer parts and trigonometry

From the second equation,

Since [x] is an integer, 8+√3−y must be an integer ⇒ y = n+√3, where n is an integer.

Hence [y] = n+1, and substituting into the first equation

As cos[sup]2[/sup]x cannot be negative or greater than 1, we must have n = 0, ±1.

Thus

Also

Putting 1 and 2 together gives

However

This is not possible as [3π] = 9. Therefore the only solution is

Last edited by JaneFairfax (2008-02-13 23:21:24)

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Integer parts and trigonometry

good work! :)