What is $tan(arcsin(12/13))$ ?

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Answer 1 · 2015-07-21T21:32:44

$tan(arcsin(12/13))=12/5$

Let $" "theta=arcsin(12/13)$

This means that we are now looking for $color(red)tantheta!$

$=>sin(theta)=12/13$

Use the identity,

$cos^2theta+sin^2theta=1$

$=>(cos^2theta+sin^2theta)/cos^2theta=1/cos^2theta$

$=>1+sin^2theta/cos^2theta=1/cos^2theta$

$=>1+tan^2theta=1/cos^2theta$

$=> tantheta=sqrt(1/cos^2(theta)-1)$

Recall : $cos^2theta= 1-sin^2theta$

$=>tantheta=sqrt(1/(1-sin^2theta)-1)$

$=>tantheta=sqrt(1/(1-(12/13)^2)-1)$

$=>tantheta=sqrt(169/(169-144)-1$

$=>tantheta=sqrt(169/25-1)$

$=>tantheta=sqrt(144/5)= 12/5$

REMEMBER what we called $theta$ was actually $arcsin(12/13)$

$=>tan(arcsin(12/13))= color(blue)(12/5)$

What is tan(arcsin(12/13)) tan(arcsin(12/13)) ?