How do you calculate $sin(tan^-1(3/4))$ ?

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Answer 1 · 2016-07-11T14:04:39

$3/5.$

If we write $tan^-1(3/4)=theta$ , then, by defn. of $tan^-1$ fun., we get, $tantheta=3/4, theta in (-pi/2,pi/2).$

Since, $tantheta >0, theta !in (-pi/2,0)$ , but, $theta in (0,pi/2)$

Now, desired value $=sin (tan^-1(3/4)) = sintheta,$ where, $tantheta=3/4.$

Now, $tantheta =3/4 rArr cottheta =4/3 rArr csc^2theta=1+cot^2theta=1+(4/3)^2=25/9 rArr csctheta=+-5/3 rArr sintheta=+-3/5.$

As, $theta in (o,pi/2), sintheta >0, so, sintheta=+3/5$

The reqd. value $=3/5.$

How do you calculate sin(tan^-1(3/4))sin(tan^-1(3/4))?