1-determiner-tan-x-2-en-fonction-de-tan-x-2-on-donne-tan-x-1-8-tan-x-2-3-la-valeur-proche-de-x-

Question Number 192924 by a.lgnaoui last updated on 31/May/23

1 ∙ determiner : \tan \frac{x}{2} en fonction de \tan x

2 ∙ on donne \tan x = \frac{1}{8} \tan \frac{x}{2} = ?

3 ∙ la valeur proche de x ?

Answered by a.lgnaoui last updated on 31/May/23

1 ∙ \tan x = \frac{2 t}{1 - t^{2}} t = \tan (\frac{x}{2})

y = \tan x \Rightarrow y (1 - t^{2}) = 2 t

y + {yt}^{2} = 2 t

t^{2} - \frac{2 t}{y} + 1 = 0 \Rightarrow {(t - \frac{1}{y})}^{2} + 1 - \frac{1}{y^{2}} = 0

{(ty - 1)}^{2} = 1 - y^{2}

t = \frac{1}{y} - \frac{\sqrt{1 - y^{2}}}{y} {\begin{cases} y \neq 0 x \neq \frac{π}{4} + k π \\ y ⩽ 1 \end{cases}

\Rightarrow \tan \frac{x}{2} = \frac{1}{\tan x} - \frac{\sqrt{1 - \tan^{2} x}}{\tan x}

2 ∙ \tan x = \frac{1}{8}

\tan \frac{x}{2} = 8 - \sqrt{63}

3 ∙ \tan \frac{x}{2} = 0, 062746 \Rightarrow \frac{x}{2} = 3, 6 °

x = 7, 2^{°}

Answered by MM42 last updated on 31/May/23

t a n x = \frac{2 t a n x}{1 - {t a n}^{2} \frac{x}{2}}

\frac{1}{8} = \frac{2 t a n \frac{x}{2}}{1 - {t a n}^{2} \frac{x}{2}} \Rightarrow {t a n}^{2} \frac{x}{2} + 16 t a n \frac{x}{2} - 1 = 0

t a n \frac{x}{2} = - 8 \pm \sqrt{65}

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