what-exact-value-sin-2x-if-given-cos-3x-2-5-

Question Number 152175 by john_santu last updated on 26/Aug/21

what exact value \sin 2 x if given

\cos 3 x = \frac{2}{\sqrt{5}}

Commented by MJS_new last updated on 26/Aug/21

\cos 3 x = {4 \cos}^{3} x - 3 \cos x

\sin 2 x = \pm 2 \cos x \sqrt{1 - \cos^{2} x}

\cos^{3} x - \frac{3}{4} \cos x - \frac{\sqrt{5}}{10} = 0

\Rightarrow \cos x = {\begin{cases} - \cos (\frac{π}{6} + \frac{1}{3} \arcsin \frac{2}{\sqrt{5}}) \\ - \sin (\frac{1}{3} \arcsin \frac{2}{\sqrt{5}}) \\ \sin (\frac{π}{3} + \frac{1}{3} \arcsin \frac{2}{\sqrt{5}}) \end{cases}

\Rightarrow

\pm 2 \cos x \sqrt{1 - \cos^{2} x} = {\begin{cases} \mp \sin (\frac{π}{3} + \frac{2}{3} \arcsin \frac{2}{\sqrt{5}}) \\ \mp \sin (\frac{2}{3} \arcsin \frac{2}{\sqrt{5}}) \\ \pm \cos (\frac{π}{6} + \frac{2}{3} \arcsin \frac{2}{\sqrt{5}}) \end{cases}

Commented by john_santu last updated on 26/Aug/21

i have try by cardano but

not solved

Commented by MJS_new last updated on 26/Aug/21

in this case we need the Trigonometric Method

Commented by MJS_new last updated on 26/Aug/21

I found no “ {nice}^{″} expressions \dots

Commented by john_santu last updated on 26/Aug/21

what the formula Trigonometry method ?

Commented by MJS_new last updated on 26/Aug/21

I posted it in comments to question 114263

Answered by mr W last updated on 26/Aug/21

\cos 6 x = 2 \times {(\frac{2}{\sqrt{5}})}^{2} - 1 = \frac{3}{5}

\sin 6 x = \pm \frac{4}{5}

6 x = 2 k π \pm \sin^{- 1} \frac{4}{5}

\Rightarrow \sin 2 x = \sin \frac{1}{3} (2 k π \pm \sin^{- 1} \frac{4}{5})

Commented by otchereabdullai@gmail.com last updated on 26/Aug/21

nice!