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Hyperbolic Functions Chapter 3. Section 6 Page 1 of 5 Section 3.6 – Hyperbolic Functions Definitions: • Recall the trig functions are related to the unit circle xy22+ =1… • In a similar way, hyperbolic functions are related to the hyperbolic function xy22−=1… • They can be expressed in terms of linear combinations of exponential growth and decay. eexx− − • sinh x = 2 eexx+ − cosh x = 2 • For the regular trig functions, sine and cosine give rise to tangent, cotangent, secant and cosecant. • In a similar way, hyperbolic sine and hyperbolic cosine give rise to… sinhx 1 1 cosh x tanhxxxx==== csch sech coth coshx sinhxxx cosh sinh C. Bellomo, revised 26-Oct-09 Chapter 3. Section 6 Page 2 of 5 Graphs of Hyperbolic Sine and Cosine: eexx− − • sinh x = 2 eexx+ − • cosh x = 2 C. Bellomo, revised 26-Oct-09 Chapter 3. Section 6 Page 3 of 5 Properties of Hyperbolic Functions: ee−−−xx−−() ee xx − • sinh(−x ) = =− =− sinh x 22 ee−−−xxxx++() ee − • cosh(−=x ) = = cosh x 22 • Q: From our definitions of even and odd, how can we classify sinh and cosh? A: Sinh is an odd function, cosh is an even function. • Q: What can we then say about their symmetry? A: Sinh is symmetric about the origin, cosh is symmetric about the y axis. • cosh22x −= sinhx 1 • And the last two properties are: sinh(x +=yxyxy ) sinh cosh + cosh sinh cosh(x +=yxyxy ) cosh cosh + sinh sinh Derivatives: ddeeee⎛⎞xx−+−− xx • sinhx ===⎜⎟ cosh x . dx dx ⎝⎠22 The derivative of sinh is cosh ddeeee⎛⎞xx+−−− xx • coshx ===⎜⎟ sinh x . dx dx ⎝⎠22 The derivative of cosh is sinh ddxxxxx⎛⎞sinh cosh (cosh )− sinh (sinh ) 1 2 • tanhx ==⎜⎟ 22 == sech x dx dx⎝⎠cosh x cosh x cosh x d • Q: Use your understanding of the chain rule and the property that cschx = 1/sinhx to find cschx . dx dd1 A: cschx = dx dxsinh x =−( 1)(sinhx )−2 cosh x −cosh x = sinh2 x =−cothxx ⋅ csch C. Bellomo, revised 26-Oct-09 Chapter 3. Section 6 Page 4 of 5 2 d • Q: Use your understanding of the property csch x = to find cschx . eexx− − dx dd2 A: csch x = dx dx exx− e− =−2( 1)(eexx −−− )2 ( ee xx + − ) . eexx+ − 2 =− eeeexxxx−−−− =−cothxx ⋅ csch d • Q: Use your understanding of the chain rule and the property that sechx = 1/coshx to find sechx . dx dd1 A: sechx = dx dxcosh x =−( 1)(coshx )−2 (sinhx ) . −sinh x = cosh2 x =−tanhxx ⋅ sech • Q: Use your understanding of the chain rule and the property that cothx = coshx/sinhx to d find cothx . dx ddxcosh A: cothx = dx dxsinh x sinhx (sinhxxx )− cosh (cosh ) = sinh2 x −1 = sinh2 x =−csch 2 x Inverse Hyperbolic Functions: • There are functions that ‘undo’ the hyperbolic functions yxyx=⇔=sinh−1 sinh yxyxy=⇔=≥cosh−1 cosh 0 yxyx=⇔=tanh−1 tanh • Because hyperbolic functions are related to exponentials, their inverses are related to logs sinh−12xxx=++ ln( 1) xR ∈ cosh−12xxxx=+−≥ ln( 1) 1 −1 11⎛⎞+ x tanhx =−<< ln⎜⎟ 1x 1 21⎝⎠− x C. Bellomo, revised 26-Oct-09 Chapter 3. Section 6 Page 5 of 5 • Let’s try to see why the first equation works… We want to find the inverse of the function yx= sinh As we usually proceed to find an inverse, we swap x and y and solve for y eeyy− − xy==sinh 2 2xe=−yy e− 21xeyy=− e2 ()exeyy2 −−= 2()10 2xx±−− 42 4(1)( 1) exxy ==±+2 1 by the quadratic formula. 2(1) So either exxy =−22 +1. But xx − +<1 0 for all x. This is not valid. or exxy =+22 +1 ⇒ y = ln( xx + + 1) • We can also differentiate any of the inverse trig functions, these formulas are in your book. C. Bellomo, revised 26-Oct-09 .
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