Patatiniuh

2022-07-13

Сompute Irrational roots of the equation ${y}^{3}-3y=\sqrt{y+2}$

zlepljalz2

Expert

$\mathrm{cos}3\theta =|\mathrm{cos}\frac{\theta }{2}|$
1. $\mathrm{cos}3\theta =\mathrm{cos}\frac{\theta }{2}$ gives
$3\theta =±\frac{\theta }{2}+{360}^{\circ }k,$
where $k\in \mathbb{Z}$, which gives $x={144}^{\circ }k$ or $x=\frac{{720}^{\circ }}{7}k.$.
Now, since $\mathrm{cos}3\theta \ge 0$, we obtain here:
$x=2\mathrm{cos}{144}^{\circ }$
or
$x=2\mathrm{cos}\frac{{720}^{\circ }}{7}.$
The case $\mathrm{cos}3\theta =-\mathrm{cos}\frac{\theta }{2}$ for you.

Do you have a similar question?