How do you simplify $sin ( sin^ -1 (-3/5) + tan^ -1(5/12))$ ?

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Answer 1 · 2016-10-17T17:30:11

$sin(sin^(-1)(-3/5)+tan^(-1)(5/12))=-16/65$

Let $sin^(-1)(-3/5)=alpha$ and $tan^(-1)(5/12)=beta$

then $sinalpha=-3/5$ and $tanbeta=5/12$

and $cosalpha=sqrt(1-(-3/5)^2)=sqrt(1-9/25)=sqrt(16/25)=4/5$

$cosbeta=1/(sqrt(1+tan^2beta))=1/(sqrt(1+(5/12)^2))=1/(sqrt(1+25/144))=1/(sqrt(169/144))=1/(13/12)=12/13$

and $sinbeta=tanbetaxxcosbeta=5/12xx1/13=5/13$

Hene, $sin(sin^(-1)(-3/5)+tan^(-1)(5/12))=sin(alpha+beta)$

= $sinalphacosbeta+cosalphasinbeta$

= $(-3/5)xx12/13+4/5xx5/13$

= $-36/65+20/65=-16/65$

How do you simplify sin ( sin^ -1 (-3/5) + tan^ -1(5/12)) sin ( sin^ -1 (-3/5) + tan^ -1(5/12)) ?