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Question Number 147492 by Kunal12588 last updated on 21/Jul/21

$$2\sqrt{19}\cos \left[\frac{1}{3}{\tan }^{-1}\left(\frac{45\sqrt{3}}{28}\right)\right]$$$$itisequalto8.How?$$

Commented by MJS_new last updated on 21/Jul/21

$$\mathrm{just}\mathrm{an}\mathrm{idea}$$$$x^3-57x-56=0$$$$x_1=-7$$$$x_2=-1$$$$x_3=8$$$$\mathrm{using}\mathrm{the}\mathrm{trigonometric}\mathrm{solution}\mathrm{formula}$$$$(\mathrm{because}p=-57\wedge q=-56\Rightarrow \frac{p^3}{27}+\frac{q^2}{4}<0\Rightarrow$$$${\mathrm{Cardano}}^{\prime }\mathrm{s}\mathrm{solution}\mathrm{method}\mathrm{is}\mathrm{not}\mathrm{possible})$$$$\mathrm{we}\mathrm{get}$$$$x_1=-2\sqrt{19}\cos (\frac{\pi }{6}+\frac{1}{3}\arcsin \frac{28\sqrt{19}}{361})$$$$x_2=-2\sqrt{19}\sin \left(\frac{1}{3}\arcsin \frac{28\sqrt{19}}{361}\right)$$$$x_3=2\sqrt{19}\sin (\frac{\pi }{3}+\frac{1}{3}\arcsin \frac{28\sqrt{19}}{361})$$$$\mathrm{now}\mathrm{we}\mathrm{must}\mathrm{show}$$$$\cos \left(\frac{1}{3}\arctan \frac{45\sqrt{3}}{28}\right)=\sin (\frac{\pi }{3}+\frac{1}{3}\arcsin \frac{28\sqrt{19}}{361})$$

Commented by Kunal12588 last updated on 22/Jul/21

$$thankyousir$$$$isthereanyothermethodthancardanowhen$$$$discriminant<0$$

Commented by MJS_new last updated on 22/Jul/21

$$\mathrm{Cardano}\mathrm{works}\mathit{o}\mathit{n}\mathit{l}\mathit{y}\mathrm{for}D⩾0$$$$\mathrm{for}D<0\mathrm{we}\mathrm{need}\mathrm{the}\mathrm{Trigonometric}\mathrm{Method}$$$$\mathrm{see}\mathrm{my}\mathrm{comments}\mathrm{to}\mathrm{question}114263$$

Answered by EDWIN88 last updated on 21/Jul/21

Commented by Kunal12588 last updated on 21/Jul/21

$$thankyousir$$

Answered by mr W last updated on 21/Jul/21

$$s=2\sqrt{19}\cos \left[\frac{1}{3}{\tan }^{-1}\left(\frac{45\sqrt{3}}{28}\right)\right]$$$$=2\sqrt{19}\cos \left[\frac{1}{3}{\cos }^{-1}\left(\frac{28}{19\sqrt{19}}\right)\right]$$$$=2\sqrt{(-\frac{-57}{3})}\cos \left[\frac{1}{3}{\cos }^{-1}(-\frac{-56}{2\sqrt{{(-\frac{-57}{3})}^3}})\right]>0$$$$whichisrootof$$$$x^2-57x-56=0$$$$(x+1)(x+7)(x-8)=0$$$$8istheonlypositiveroot,therefore$$$$2\sqrt{19}\cos \left[\frac{1}{3}{\tan }^{-1}\left(\frac{45\sqrt{3}}{28}\right)\right]=8$$

Commented by Kunal12588 last updated on 22/Jul/21

$$thankyousir$$

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