The roots of the equation

2009-02-21 9:51 pm

if
cot7a
= 1-21tan^2a+35tab^4a-7tan^6a/(7tana-35tan^3a+21tan^5a-tan^7a)

for cot7a=0 , cos7a = 2n兀+/- 兀/2 where n is any integer

=> 1-21tan^2a+35tab^4a-7tan^6a = 0

show that

tan^2(兀/14) , tan^2(3兀/14) and tan^2(5兀/14) are the roots of

7x^3-35x^2+21x-1=0

更新1:

why?? cot7a = 0 => cot na/7 where n = 1,....,6

更新2:

sub a = pi/7, 2pi/7, ... , 6pi/7 into cot7a=0 It is undefined!

回答 (1)

Prof. Physics

2009-02-22 12:48 am

✔ 最佳答案

Note that this question is very similar to 2007 AL pure maths paper 1 Q. 11, you may look that for reference.

For detail solution, please go to:

http://i726.photobucket.com/albums/ww265/physicsworld2010/physicsworld01Feb211647.jpg

2009-02-22 13:16:31 補充：
It is because sub a = pi/7, 2pi/7, ... , 6pi/7, cot7a = 0
So, cot7a = 0, cotna/7 where n = 1, ... , 6
You may try it.

2009-02-22 20:33:44 補充：
Hey, you are putting a = pi/7, 2pi/7 into tan7a, that's why it is undefined.

In fact, it is infinity.

If you have the reciprocal, that is cot7a, that is 1 / infinity, you have 0.

Try it

參考: Physics king

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