Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I know one root of the equation (eq.1),

$x^4+ax^3+2x^2-ax+1 = 0$

is,

$x_1 = \tan\big(\tfrac{1}{4}\arcsin(\tfrac{4}{a})\big)$

How to find the other three roots of eq.1 expressed similarly in terms of trigonometric and/or inverse trigonometric functions?

share|improve this question
1  
There are three non-principal values of the arcsine that have different tangents when divided by 4 -- namely $\theta+2\pi$, $\pi-\theta$ and $3\pi-\theta$, where $\theta$ is the principal arcsine. If $\tan(\theta/4)$ is a root, chances are excellent that the others will be, too. –  Henning Makholm Feb 11 '12 at 18:00
    
Thanks for the tip. I found that, given $x = \tan(\theta/4)$, then two roots are given by $\theta = \pi/2 +\arcsin\big(\pm\sqrt{1-(4/a)^2}\big)$, and the other two are $\theta = -3\pi/2 +\arcsin\big(\pm\sqrt{1-(4/a)^2}\big)$. (I forgot to mention that a is non-zero.) –  Tito Piezas III Feb 11 '12 at 19:06

1 Answer 1

up vote 1 down vote accepted

I wanted to find a clever solution in terms of the tangent being the slope of some line crossing an algebraic curve... no luck. This is what I got instead: $$\begin{split}x^4+2x^2+1&=a(x-x^3)\\ (x^2+1)^2&=a(x-x^3)\end{split}\tag1$$ Now substitution $x=\tan t$ is natural on the left, because $x^2+1=1/\cos^2 t$. $$\begin{split}\frac{1}{\cos^4 t}&=a(\tan t-\tan^3 t)\\ 1 &=a(\sin t\cos^3t-\sin^3 t\cos t) \end{split}\tag2$$ The rest flows easily: factor out $\sin t \cos t$ and turn it into $\frac12 \sin 2t$, then use $\cos^2 t-\sin^2 t=\cos 2t$, finally arriving at $$1=\frac{a}{4}\sin 4t \tag3$$ Now it's time to pay attention to domains: the substitution $x=\tan t$ is a bijection between $\mathbb R$ and $(-\pi/2,\pi/2)$. In the interval $(-\pi/2,\pi/2)$, which is two periods of the function $\sin 4t$, equation (3) has

  • no roots if $|a|<4$
  • two (double) roots if $|a|=4$
  • four roots if $|a|>4$

It is awkward to write down the roots keeping them all in $(-\pi/2,\pi/2)$. Since all we care about is $\tan t$, adding or subtracting a multiple of $\pi$ is acceptable. For example: $\theta$, $\theta+\pi/2$, $\pi/4-\theta$, and $3\pi/4-\theta$, where $\theta=\arcsin (4/a)$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.