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Question-111079




Question Number 111079 by shahria14 last updated on 02/Sep/20
Answered by mr W last updated on 02/Sep/20
t=cos^(−1) (5/(13))  ⇒cos t=(5/(13))  ⇒sin t=((12)/(13))  tan (t/2)=((sin t)/(1+cos t))=(((12)/(13))/(1+(5/(13))))=(2/3)  ⇒(1/2)cos^(−1) (5/(13))=(t/2)=tan^(−1) (2/3)
$${t}=\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{5}}{\mathrm{13}} \\ $$$$\Rightarrow\mathrm{cos}\:{t}=\frac{\mathrm{5}}{\mathrm{13}} \\ $$$$\Rightarrow\mathrm{sin}\:{t}=\frac{\mathrm{12}}{\mathrm{13}} \\ $$$$\mathrm{tan}\:\frac{{t}}{\mathrm{2}}=\frac{\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{cos}\:{t}}=\frac{\frac{\mathrm{12}}{\mathrm{13}}}{\mathrm{1}+\frac{\mathrm{5}}{\mathrm{13}}}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{5}}{\mathrm{13}}=\frac{{t}}{\mathrm{2}}=\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{2}}{\mathrm{3}} \\ $$

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