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

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

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Answer 1 · 2016-09-11T05:57:48

$sin (arctan (\frac{3}{4})) = \pm \frac{3}{5}$ $sin(arctan(3/4))=+-3/5$

Explanation:

$sin (arctan (\frac{3}{4}))$ $sin(arctan(3/4))$ literally means sine of an angle whose tangent is $\frac{3}{4}$ $3/4$ . Let the angle be $θ = arctan (\frac{3}{4})$ $theta=arctan(3/4)$

As $tan θ = \frac{3}{4}$ $tantheta=3/4$ , $cot θ = \frac{4}{3}$ $cottheta=4/3$ (as it is reciprocal)

Hence $csc \cdot 2 θ = 1 + {cot}^{2} θ = 1 + \frac{16}{9} = \frac{25}{9}$ $csc*2theta=1+cot^2theta=1+16/9=25/9$

Hence $csc θ = \pm \frac{5}{3}$ $csctheta=+-5/3$ and $sin θ = \pm \frac{3}{5}$ $sintheta=+-3/5$

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

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

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