E-StoreHome » Ask & Discuss » Mathematics. » Trignometry « Back to Discussion
Trignometry
11 Mar 2008 10:10:47 IST
0 People liked this
3
381
) somewhereComments (3)
11 Mar 2008 11:06:31 IST
Like
0 people liked this
very nice work, anchit
The "official" solution used the same idea as Anchit's but simplified substantially. This is not to take credit away from Anchit, his is really a great effort, kudos.
tan
is rational iff tan
is rational where = -
and 2 =
is rational iff tan is rational where = - and 2 = This means tan2 is rational.
is rational.tan2 = 2tan/(1-tan2) = p/q
= 2tan/(1-tan2) = p/qHence ptan2 + 2qtan - p =0
+ 2qtan - p =0Tan is therefore rational if D = 4(p2+q2) is a perfect square which means p2+q2 is a perfect square
is therefore rational if D = 4(p2+q2) is a perfect square which means p2+q2 is a perfect squareFree Sign Up!
Preparing for IIT-JEE ?
Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor
@ INR 1,995/-
For Quick Info
Find Posts by Topics
Physics.
Topics
9396
20092
3672
2186
7617
2827
2638
Mathematics.
Chemistry.
Biology
Institutes
Parents
Board
Fun Zone
|
|
|
|
|
| 35,229,755 Engineering Questions Attempted |
6,69,530 Visitors Monthly |
79 Engg. Exam Covered |
2114 Students Currently Online |
Connect with Us  |
 |
Vriti Communities
Privacy Policy © Vriti 2011
=(3pq2 - p3) / (q3- 3 p2q) (on solving)
2 tan beta (q3- 3 p2q) = ( 1 - tan2 beta)(3pq2 - p3)
on solving this gives---
tan2 beta( p3- 3pq2) + tan beta ( 6 p2q- 2q3) + (3pq2 - p3) =0
its Discriminant is--
( 6 p2q- 2q3) 2 - 4( p3- 3pq2)(3pq2 - p3)
= 4((3p2q - q3)2 + 4( p3- 3pq2)2
=4 (q2(9p4 + q4 - 6p2q2) + p2(p4 + 9q4 - 6p2q2))
=4 ( 9p4q2 + q6 - 6p2q4 + p6 + 9q4p2 -6p4q2)
=4 ( p6 + q6 + 3p2q2(p2+q2))
=4((p2+q2) (p4+q4 + 2p2q2)
[on expanding p^6+q^6 as (p^2)^3 +(q^2)^3 ]
=(2(p2+q2))2(p2+q2)
hence
tan beta would be rational if D is a perfect square which is possible if
(p2+q2) is a perfect square