If $${\sin ^{ - 1}}{\alpha \over {17}} + {\cos ^{ - 1}}{4 \over 5} - {\tan ^{ - 1}}{{77} \over {36}} = 0,0 < \alpha < 13$$, then ${\sin ^{ - 1}}(\sin \alpha ) + {\cos ^{ - 1}}(\cos \alpha )$ is equal to :
Solution
$\sin ^{-1}\left(\frac{\alpha}{17}\right)=-\cos ^{4}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{77}{36}\right)$
<br/><br/>Let $\cos ^{-1}\left(\frac{4}{5}\right)=p$ and $\tan ^{-1}\left(\frac{77}{36}\right)=q$
<br/><br/>$\Rightarrow \sin \left(\sin ^{-1} \frac{\alpha}{17}\right)=\sin (q-p)$
<br/><br/>$=\sin q \cdot \cos p-\cos q \cdot \sin p$
<br/><br/>$\Rightarrow \frac{\alpha}{17}=\frac{77}{85} \cdot \frac{4}{5}-\frac{36}{85} \cdot \frac{3}{5}$
<br/><br/>$\Rightarrow \alpha=\frac{200}{25}=8$
<br/><br/>$\sin ^{-1} \sin 8+\cos ^{-1} \cos 8$
<br/><br/>$= -8+3 \pi+8-2 \pi$
<br/><br/>$=\pi$
About this question
Subject: Mathematics · Chapter: Inverse Trigonometric Functions · Topic: Domain and Range
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