prove-that-2sin-1-3-4-tan-1-24-7-

Question Number 1206 by gs8763330@gmail.com last updated on 14/Jul/15

p r o v e t h a t {2 \sin}^{- 1} 3 / 4 = \tan^{- 1} 24 / 7

Commented by 123456 last updated on 14/Jul/15

\sin α = a \overset{b . c}{\Leftrightarrow} α = \sin^{- 1} a

\cos α = b \overset{b . c}{\Leftrightarrow} α = \cos^{- 1} b

\tan α = \frac{\sin α}{\cos α} = t \overset{b . c}{\Leftrightarrow} α = \tan^{- 1} \frac{\sin α}{\cos α} = \tan^{- 1} \frac{a}{b} = \tan^{- 1} t

Answered by prakash jain last updated on 14/Jul/15

\sin^{- 1} \frac{3}{5} = θ

\sin θ = \frac{3}{5}, \cos θ = \frac{4}{5}

\tan 2 θ = \frac{2 \sin θ \cos θ}{\cos^{2} θ - \sin^{2} θ} = \frac{2 \cdot \frac{3}{5} \cdot \frac{4}{5}}{\frac{16}{25} - \frac{9}{25}} = \frac{24}{7}

\Rightarrow 2 θ = \tan^{- 1} \frac{24}{7}

{2 \sin}^{- 1} \frac{3}{5} = \tan^{- 1} \frac{24}{7}

There is a typo in your question . It

should be

{2 \sin}^{- 1} \frac{3}{5} = \tan^{- 1} \frac{24}{7}