How do you evaluate #sin(arctan(3/4))# without a calculator?

1 Answer
Shwetank Mauria
Sep 11, 2016

#sin(arctan(3/4))=+-3/5#

Explanation:

#sin(arctan(3/4))# literally means sine of an angle whose tangent is #3/4#. Let the angle be #theta=arctan(3/4)#

As #tantheta=3/4#, #cottheta=4/3# (as it is reciprocal)

Hence #csc*2theta=1+cot^2theta=1+16/9=25/9#

Hence #csctheta=+-5/3# and #sintheta=+-3/5#

Note that as #tantheta# is positive, it is in firs or third quadrant and hence, we can have #sintheta# positive as well as negative.

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