[san_M]

Show that tan 9 is the smallest positive root of the equation t^4 - 4t^3 - 14t^2 -4t + 1 = 0.

I used DMT for tan 5x and obtained (t-1)(t^4 - 4t^3 - 14t^2 -4t + 1) as the polynomial.

I then equate tan5x=1 and obtained x = 9, 45, 81, 117, 153 with tan9, tan 45, tan 81, tan 117, tan 153 as the following roots.

However I'm unsure what to do with the t-1 since I've been asked to show tan 9 is a root of the quartic. My teacher wrote down tanx is not equal to one, but I don't understand why. Can anyone explain please?

Thank you

[Ree69]

Haha, this is almost identical to my thread here.

You've done all the hard work (I actually read this to solve mine).



(After expressing in terms of from de Moivre's).

As you said, . These are the five solutions to the above equation. So when doing , we know that or . If . So the other four answers must be the solutions to .

[san_M]

(Original post by Ree69)
Haha, this is almost identical to my thread here.

You've done all the hard work (I actually read this to solve mine).



(After expressing in terms of from de Moivre's).

As you said, . These are the five solutions to the above equation. So when doing , we know that or . If . So the other four answers must be the solutions to .

aha! you helped clarify my problems as well! was doing this question pretty late and was tired. But it's clear now thanks glad I helped you in return too