Hyperbolic functions:

Question

!!"sech"^-1 (sin theta)!! is?

  • !!log(cot(theta/2))!!
  • !!log(tan(theta/2))!!
  • !!log(cot(theta))!!
  • !!log(tan(theta))!!

Solution

!!coshx=(e^x+e^-x)/2!!

!!=>cosh^-1x=ln(x+sqrt(x^2-1))!!

!!sech^-1 sintheta =cosh^-1(1/sintheta)!! since !!sech^-1 a =cosh^-1(1/a)!!

!!cosh^-1(1/sintheta)=ln(1/sintheta+sqrt(1-sin^2theta)/sintheta)ln(1/sintheta+(costheta)/sintheta)!!

!!ln((1+costheta)/sintheta)=ln((2cos^2theta/2)/(2sin(theta/2 )cos(theta/2)))=log(cot(theta/2))!!

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