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  • Find the simplified from of cos inverse into bracket 3 divided 5 cos x plus 4 divided 5 sin x where x belongs to into square bracket minus 3pi divided 4 pi divided 4

Find the simplified from of \cos^{-1}\left ( \frac{3}{5}\cos x +\frac{4}{5}\sin x \right ) , where x\epsilon \left [ \frac{-3\pi}{4},\frac{\pi}{4} \right ]

Answers (1)

Let \cos y=\frac{3}{5}

\Rightarrow \sin y=\frac{4}{5}

\Rightarrow y=\cos^{-1}\frac{3}{5}=\sin^{-1}\frac{4}{5}=\tan^{-1}\frac{4}{3}

\therefore \cos^{-1}\left [ \cos y. \cos x+\sin y. \sin x \right ]

\left [ since, \cos\left ( A-B \right ) = \cos A.\cos B + \sin A. \sin B \right ]

=\cos^{-1}\left [ \cos\left ( y-x \right ) \right ]

\left [ scine, \cos \left ( \cos^{-1}x \right )=x,x\epsilon \left \{ -1,1 \right \} \right ]

=y-x

\left [ scine, y=\tan^{-1}\frac{4}{3} \right ]

=\tan^{-1}\frac{4}{3} -x

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