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

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Answer 1 · 2016-09-11T05:57:48

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

$sin(arctan(3/4))$ 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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