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Simplify \(\sin \left( \sin^{-1} \frac{3}{5} + \tan^{-1} 2 \right).\)

 Apr 12, 2020
 #1
avatar+9472 
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Simplify \(\sin \left( \sin^{-1} \frac{3}{5} + \tan^{-1} 2 \right). \)

 

Hello Guest!

 

\(sin(\frac{1}{sin(\frac{3}{5})}+\frac{1}{tan(2)})\\ =sin(csc(\frac{3}{5})+\frac{1}{tan(2)})\\ =sin(\pm \frac{\sqrt{1+tan^2(\frac{3}{5})}}{tan(\frac{3}{5})}+\frac{1}{tan(2)})\)

I dont get any further.

crying  !

 Apr 12, 2020
 #2
avatar+7825 
0

Assuming \(\sin^{-1} x \) and \(\tan^{-1} x \) means inverse trigonometric functions,

 

\(\sin\left(\sin^{-1} \dfrac35 + \tan^{-1} 2\right)\\ = \sin\left(\tan^{-1} \dfrac34 + \tan^{-1}2 \right)\\ = \sin \tan^{-1} \dfrac34 \cos \tan^{-1} 2 + \cos \tan^{-1} \dfrac34 \sin \tan^{-1} 2\\ = \dfrac35 \dfrac1{\sqrt5} + \dfrac45\dfrac{2}{\sqrt5}\\ = \dfrac{11}{5\sqrt5}\)

.
 Apr 14, 2020

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