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

1 Answer
Jul 11, 2016

#3/5.#

Explanation:

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.#