MTH 252 Integral Calculus Chapter 7 Applications of the Definite Integral Section 7 9 Hyperbolic Functions and Hanging Cables Copyright 2005 by Ron Wallace all rights reserved Reminder The Circular Functions cos x a a b x sin x b 2 2 cos x sin x 1 x 1 Unit Circle a2 b2 1 sin x tan x cos x 1 cot x tan x 1 sec x cos x 1 csc x sin x Euler s Formula ix e cos x i sin x NOTE This will be proved in MTH 253 This identity implies that eix e ix cos x 2 eix e ix sin x 2i What functions do you get if you remove the i s Hyperbolic Functions e x e x cosh x 2 e x e x sinh x 2 cosh 2 x sinh 2 x 1 If a coshx and b sinhx then any ordered pair a b will be a point on the unit hyperbola a2 b2 1 Also x e cosh x sinh x Hyperbolic Functions cosh x e x e x 2 2 2 cosh x sinh x 1 sinh x e x e x 2 tanh x sinh x cosh x 2 2 1 tanh x sech x sech x 1 cosh x coth x 1 tanh x csch x 1 sinh x 2 2 coth x 1 csch x Hyperbolic Functions Derivatives d cosh x sinh x dx d sinh x cosh x dx d tanh x sech 2 x dx d sech x sech x tanh x dx d coth x csch 2 x dx d csch x csch x coth x dx Inverse Hyperbolic Functions y cosh 1 x d 1 1 cosh x dx x2 1 y sinh 1 x See page 512 formula 25 d 1 1 sinh x dx x 2 1 y tanh 1 x d 1 1 tanh x dx 1 x2 x cosh y x sinh y See page 512 formula 24 x tanh y See page 512 formula 26