Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 114108 by bemath last updated on 17/Sep/20

$$provethat2{\tan }^{-1}\left(\frac{2}{3}\right)={\sin }^{-1}\left(\frac{12}{13}\right)$$

Answered by bobhans last updated on 17/Sep/20

$$let{\tan }^{-1}\left(\frac{2}{3}\right)=x\to \tan x=\frac{2}{3}$$$$and\{\begin{array}{l}\sin x=\frac{2}{\sqrt{13}}\\ \cos x=\frac{3}{\sqrt{13}}\end{array}$$$$\iff 2x={\sin }^{-1}\left(\frac{12}{13}\right)$$$$\iff \sin \left(2x\right)=\sin \left({\sin }^{-1}\left(\frac{12}{13}\right)\right)$$$$\Rightarrow 2\sin x\cos x=\frac{12}{13}$$$$\Rightarrow 2\left(\frac{2}{\sqrt{13}}\right)\left(\frac{3}{\sqrt{13}}\right)=\frac{12}{13}$$

Answered by Dwaipayan Shikari last updated on 17/Sep/20

$${tan}^{-1}\left(\frac{\frac{2}{3}+\frac{2}{3}}{1-\frac{4}{9}}\right)={tan}^{-1}\frac{12}{5}$$$$tan\theta =\frac{12}{5}$$$$sec\theta =\frac{13}{5}$$$$cos\theta =\frac{5}{13}andsin\theta =\sqrt{1-\frac{5^2}{{13}^2}}=\frac{12}{13}$$$$\theta ={sin}^{-1}\frac{12}{13}$$$$\theta =2{tan}^{-1}\frac{2}{3}\left(Whichistrue\right)$$

Answered by physicstutes last updated on 17/Sep/20

$$\mathrm{if}\sin \theta =\frac{12}{13},\mathrm{then}\tan \theta =\frac{12}{5}$$$$\tan \left(\frac{\theta }{2}\right)=\frac{\sin \theta }{1+\cos \theta }$$$$\cos \theta =\frac{5}{13}\Rightarrow \tan \left(\frac{\theta }{2}\right)=\frac{\frac{12}{13}}{1+\frac{5}{13}}=\frac{12}{18}=\frac{2}{3}$$$$\mathrm{hence}\tan \left(\frac{\theta }{2}\right)=\frac{2}{3}\Rightarrow \theta =2{\tan }^{-1}\left(\frac{2}{3}\right)$$$$\mathrm{thus},\theta ={\sin }^{-1}\left(\frac{12}{13}\right)=2{\tan }^{-1}\left(\frac{2}{3}\right)$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com