Question

If 270° <teta< 360°and cos teta=1/4 find teta/2​

Answer 1

SOLUTION :-

GIVEN :-

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

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

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

Now

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

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

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

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

$\because \: \: \displaystyle \sf{ {270}^{ \circ} &lt; \theta &lt; {360}^{ \circ} \: \: }$

$\therefore \: \: \displaystyle \sf{ {135}^{ \circ} &lt; \frac{ \theta}{2} &lt; {180}^{ \circ} \: \: }$

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

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

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

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

FINAL ANSWER :-

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

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Find the Value of

3 + cot 80 cot 20/cot80+cot20

https://brainly.in/question/17024513

2. Prove that :

tan9A - tan6A -tan3A = tan9A.tan 6A.tan 3A

https://brainly.in/question/9394506