Math, asked by madhur65, 6 months ago

If cot theta/2 =2.5 then find cosec theta.

Answers

Answered by BrainlyPopularman
6

GIVEN :

  \\ \implies \bf \cot \left(\dfrac{ \theta}{2}\right) = 2.5 \\

TO FIND :

  \\ \implies \bf cosec(\theta) =  \: ? \\

SOLUTION :

  \\ \implies \bf \cot \left(\dfrac{ \theta}{2}\right) = 2.5 \\

  \\ \implies \bf \tan \left(\dfrac{ \theta}{2}\right) =  \dfrac{1}{2.5 }\\

  \\ \implies \bf \tan \left(\dfrac{ \theta}{2}\right) =  \dfrac{2}{5}\\

• We know that –

  \\ \implies \pink{ \boxed{ \bf \tan( \theta)=  \dfrac{2\tan \left(\dfrac{ \theta}{2}\right)}{1 - \tan^{2}  \left(\dfrac{ \theta}{2}\right)}}}\\

• Put the values –

  \\ \implies\bf \tan( \theta)=  \dfrac{2\left(\dfrac{2}{5} \right)}{1 - \left(\dfrac{2}{5} \right)^{2}}\\

  \\ \implies\bf \tan( \theta)=  \dfrac{\dfrac{4}{5}}{1 -\dfrac{4}{25}}\\

  \\ \implies\bf \tan( \theta)=  \dfrac{\dfrac{4}{5}}{\dfrac{25 - 4}{25}}\\

  \\ \implies\bf \tan( \theta)=  \dfrac{\dfrac{4}{5}}{\dfrac{21}{25}}\\

  \\ \implies\bf \tan( \theta)= \dfrac{4}{5} \times \dfrac{25}{21}\\

  \\ \implies\bf \tan( \theta)= 4\times \dfrac{5}{21}\\

  \\ \implies\bf \tan( \theta)= \dfrac{20}{21}\\

• We should write this as –

  \\ \implies\bf  \dfrac{ \sin\theta}{ \cos\theta}= \dfrac{20}{21}\\

  \\ \implies\bf  \dfrac{ \sin\theta}{ \sqrt{1 -\sin^{2} \theta}}= \dfrac{20}{21}\\

  \\ \implies\bf  \dfrac{ \sin^{2} \theta}{ {1 -\sin^{2} \theta}}= \dfrac{400}{441}\\

  \\ \implies\bf441\sin^{2} \theta = 400(1 -\sin^{2} \theta)\\

  \\ \implies\bf441\sin^{2} \theta = 400 -400\sin^{2} \theta\\

  \\ \implies\bf441\sin^{2} \theta + 400\sin^{2} \theta= 400\\

  \\ \implies\bf841\sin^{2} \theta= 400\\

  \\ \implies\bf\sin^{2} \theta=  \dfrac{400}{841}\\

  \\ \implies\bf\sin\theta= \pm\sqrt \dfrac{400}{841}\\

  \\ \implies\bf\sin\theta=\pm \dfrac{20}{29}\\

  \\ \implies\bf \dfrac{1}{\sin\theta}=\pm \dfrac{29}{20}\\

  \\ \implies \large{ \boxed{\bf cosec(\theta)= \pm \dfrac{29}{20}}}\\

Answered by prabhas24480
0

GIVEN :–

  \\ \implies \bf \cot \left(\dfrac{ \theta}{2}\right) = 2.5 \\

TO FIND :–

  \\ \implies \bf cosec(\theta) =  \: ? \\

SOLUTION :–

  \\ \implies \bf \cot \left(\dfrac{ \theta}{2}\right) = 2.5 \\

  \\ \implies \bf \tan \left(\dfrac{ \theta}{2}\right) =  \dfrac{1}{2.5 }\\

  \\ \implies \bf \tan \left(\dfrac{ \theta}{2}\right) =  \dfrac{2}{5}\\

• We know that –

  \\ \implies \pink{ \boxed{ \bf \tan( \theta)=  \dfrac{2\tan \left(\dfrac{ \theta}{2}\right)}{1 - \tan^{2}  \left(\dfrac{ \theta}{2}\right)}}}\\

• Put the values –

  \\ \implies\bf \tan( \theta)=  \dfrac{2\left(\dfrac{2}{5} \right)}{1 - \left(\dfrac{2}{5} \right)^{2}}\\

  \\ \implies\bf \tan( \theta)=  \dfrac{\dfrac{4}{5}}{1 -\dfrac{4}{25}}\\

  \\ \implies\bf \tan( \theta)=  \dfrac{\dfrac{4}{5}}{\dfrac{25 - 4}{25}}\\

  \\ \implies\bf \tan( \theta)=  \dfrac{\dfrac{4}{5}}{\dfrac{21}{25}}\\

  \\ \implies\bf \tan( \theta)= \dfrac{4}{5} \times \dfrac{25}{21}\\

  \\ \implies\bf \tan( \theta)= 4\times \dfrac{5}{21}\\

  \\ \implies\bf \tan( \theta)= \dfrac{20}{21}\\

• We should write this as –

  \\ \implies\bf  \dfrac{ \sin\theta}{ \cos\theta}= \dfrac{20}{21}\\

  \\ \implies\bf  \dfrac{ \sin\theta}{ \sqrt{1 -\sin^{2} \theta}}= \dfrac{20}{21}\\

  \\ \implies\bf  \dfrac{ \sin^{2} \theta}{ {1 -\sin^{2} \theta}}= \dfrac{400}{441}\\

  \\ \implies\bf441\sin^{2} \theta = 400(1 -\sin^{2} \theta)\\

  \\ \implies\bf441\sin^{2} \theta = 400 -400\sin^{2} \theta\\

  \\ \implies\bf441\sin^{2} \theta + 400\sin^{2} \theta= 400\\

  \\ \implies\bf841\sin^{2} \theta= 400\\

  \\ \implies\bf\sin^{2} \theta=  \dfrac{400}{841}\\

  \\ \implies\bf\sin\theta= \pm\sqrt \dfrac{400}{841}\\

  \\ \implies\bf\sin\theta=\pm \dfrac{20}{29}\\

  \\ \implies\bf \dfrac{1}{\sin\theta}=\pm \dfrac{29}{20}\\

  \\ \implies \large{ \boxed{\bf cosec(\theta)= \pm \dfrac{29}{20}}}\\

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