Question

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

Answer 1

`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.`