Question

if 270 is less than theta is less than 360 and cos theta is 1/4 find tan theta /2 ​

Answer 1

SOLUTION :-

GIVEN :-

$\displaystyle \sf{ {270}^{ \circ} < \theta < {360}^{ \circ} \: \: \: and \: \cos \theta = \frac{1}{4} }$

TO DETERMINE :-

$\displaystyle \sf{ \tan \frac{ \theta}{2} }$

FORMULA TO BE IMPLEMENTED :-

We are aware of the Trigonometric identity that

$\displaystyle \sf{ {\tan}^{2} \frac{ \theta}{2} = \frac{1 - \cos \theta}{1 + \cos \theta} }$

EVALUATION :-

Here it is given that

$\displaystyle \sf{ {270}^{ \circ} < \theta < {360}^{ \circ} \: \: \: and \: \cos \theta = \frac{1}{4} }$

Now

$\displaystyle \sf{ {\tan}^{2} \frac{ \theta}{2} = \frac{1 - \cos \theta}{1 + \cos \theta} }$

$\implies \displaystyle \sf{ {\tan}^{2} \frac{ \theta}{2} = \frac{1 - \frac{1}{4} }{1 + \frac{1}{4} } }$

$\implies \displaystyle \sf{ {\tan}^{2} \frac{ \theta}{2} = \frac{ \frac{4 - 1}{4} }{ \frac{4 + 1}{4} } }$

$\implies \displaystyle \sf{ {\tan}^{2} \frac{ \theta}{2} = \frac{3 }{5} }$

$\because \: \: \displaystyle \sf{ {270}^{ \circ} < \theta < {360}^{ \circ} \: \: }$

$\therefore \: \: \displaystyle \sf{ {135}^{ \circ} < \frac{ \theta}{2} < {180}^{ \circ} \: \: }$

$\therefore \: \: \displaystyle \sf{ \frac{ \theta}{2} \: \: lies \: in \: second \: quadrant }$

$\therefore \: \: \displaystyle \sf{ \tan \frac{ \theta}{2} \: \: \: is \: negative }$

$\therefore \: \: \displaystyle \sf{ {\tan}^{2} \frac{ \theta}{2} = \frac{3 }{5} } \: \: gives$

$\displaystyle \sf{ {\tan} \frac{ \theta}{2} = - \sqrt{ \frac{3 }{5} }}$

FINAL ANSWER :-

$\boxed{\displaystyle \sf{ \: \: {\tan} \frac{ \theta}{2} = - \sqrt{ \frac{3 }{5} }} \: \: \: }$

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