Asked by Rosco
if sinh x=tan y, show that x=In(secy+tany)
Answers
Answered by
Steve
sinh x = (e^x - e^-x)/2
So, now you have
e^x - e^-x = 2tan y
e^2x - 1 = e^x * 2tan y
e^2x - 2tany e^x - 1 = 0
Now use the quadratic formula to get
e^x = [2tany ±√(4tan^2y+4)]/2
= tany±√(tan^2y+1)
= tany±secy
x = ln(tany±secy)
Pick the principal branch of the curve.
So, now you have
e^x - e^-x = 2tan y
e^2x - 1 = e^x * 2tan y
e^2x - 2tany e^x - 1 = 0
Now use the quadratic formula to get
e^x = [2tany ±√(4tan^2y+4)]/2
= tany±√(tan^2y+1)
= tany±secy
x = ln(tany±secy)
Pick the principal branch of the curve.
Answered by
Rosco
if a=c cosh x and b=c sinh x, prove that(a+b)^2e^-2x=a^2-b^2
Answered by
Steve
(a+b)^2e^-2x=a^2-b^2
divide both sides by a+b:
(a+b) e^-2x = a-b
(a+b)/(a-b) = e^2x
a+b = ce^x
a-b = ce^-x
and the rest follows.
divide both sides by a+b:
(a+b) e^-2x = a-b
(a+b)/(a-b) = e^2x
a+b = ce^x
a-b = ce^-x
and the rest follows.