How do you evaluate #tan(csc^-1(-2))# without a calculator?

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Answer 1 · 2016-08-19T07:11:18

#tan(csc^-1(sqrt(7)/2))=1/sqrt3#

Let #theta=csc^-1(-2)#

or

#csctheta=-2#

or

#1/sintheta=-2#

or

#sintheta=-1/2#

or

#theta=sin^-1(-1/2)#

Hence

#p/h=1/2# (Since #theta# lies in third quadrant; hence #-# sign not taken in consideration)

#costheta=b/h#

or

#costheta=sqrt(h^2-p^2)/h#

or

#costheta=sqrt((2^2-1^2))/2#

or

#costheta=sqrt(4-1)/2#

or

#costheta=sqrt3/2#

Therefore
#tantheta=sintheta/costheta#

or

#tan theta=(1/2)/((sqrt3)/2#

or

#tan theta=1/sqrt3#

or

#tan(csc^-1(sqrt(7)/2))=1/sqrt3#