Let#" "theta=arcsin(-5/13)#
This means that we are now looking for #color(red)cottheta!#
#=>sin(theta)=-5/13#
Use the identity,
#cos^2theta+sin^2theta=1#
NB : #sintheta# is negative so #theta# is also negative.
We shall the importance of this info later.
#=>(cos^2theta+sin^2theta)/sin^2theta=1/sin^2theta#
#=>cos^2theta/sin^2theta+1=1/sin^2theta#
#=>cot^2theta+1=1/sin^2theta#
#=>cot^2theta=1/sin^2x-1#
#=> cottheta=+-sqrt(1/sin^2(theta)-1)#
#=>cottheta=+-sqrt(1/(-5/13)^2-1)=+-sqrt(169/25-1)=+-sqrt(144/25)=+-12/5#
WE saw the evidence previously that #theta# should be negative only.
And since #cottheta# is odd #=>cott(-A)=-cot(A)# Where #A# is a positive angle.
So, it becomes clear that #cottheta=color(blue)+12/5#
REMEMBER what we called #theta# was actually #arcsin(-15/13)#
#=>cot(arcsin(-5/13)) = color(blue)(12/5)#
