Evaluate : tan {(1/2)cos^(−1) ((√5)/3)} ?


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Question Number 51356 by rahul 19 last updated on 26/Dec/18
$Evaluate : tan {(1/2)cos^(−1) ((√5)/3)} ?$
$E v a l u a t e :$ $\tan {\frac{1}{2} \cos^{- 1} \frac{\sqrt{5}}{3}} ?$
Commented by rahul 19 last updated on 26/Dec/18

$A n s \to \frac{3 - \sqrt{5}}{2} .$
	Answered by mr W last updated on 26/Dec/18
	$let c=cos^(−1) ((√5)/3) ⇒cos c=((√5)/3) tan {(1/2)cos^(−1) ((√5)/3)}=tan (c/2)=(√((1−cos c)/(1+cos c))) =(√((1−((√5)/3))/(1+((√5)/3))))=(√((3−(√5))/(3+(√5))))=(√(((3−(√5))^2 )/((3+(√5))(3−(√5))))) =((3−(√5))/2)$
	$l e t c = \cos^{- 1} \frac{\sqrt{5}}{3}$ $\Rightarrow \cos c = \frac{\sqrt{5}}{3}$ $\tan {\frac{1}{2} \cos^{- 1} \frac{\sqrt{5}}{3}} = \tan \frac{c}{2} = \sqrt{\frac{1 - \cos c}{1 + \cos c}}$ $= \sqrt{\frac{1 - \frac{\sqrt{5}}{3}}{1 + \frac{\sqrt{5}}{3}}} = \sqrt{\frac{3 - \sqrt{5}}{3 + \sqrt{5}}} = \sqrt{\frac{{(3 - \sqrt{5})}^{2}}{(3 + \sqrt{5}) (3 - \sqrt{5})}}$ $= \frac{3 - \sqrt{5}}{2}$
	Commented by rahul 19 last updated on 26/Dec/18

	$t h a n k y o u s i r!$ $D e f i n i t e l y s h o r t e r t h a n t h e p r e v i o u s$ $o n e! :)$
	Answered by ajfour last updated on 26/Dec/18

	$\tan [\frac{1}{2} (2 θ)] = \tan [\frac{1}{2} \sin^{- 1} \frac{2}{3}]$ $\Rightarrow \sin 2 θ = \frac{2}{3} = \frac{2 t}{1 + t^{2}} \forall t = \tan θ$ $\Rightarrow t^{2} - 3 t + 1 = 0$ $t = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2} .$
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