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Cosh 2 x Cosh2 x 1 Sinh 2 x Cosh2 x 1
2-4- Hyperbolic functions : Hyperbolic functions are used to describe the motions of waves in elastic solids ; the shapes of electric power lines ; temperature distributions in metal fins that cool pipes …etc. The hyperbolic sine (Sinh) and hyperbolic cosine (Cosh) are defined by the following equations : 1. Sinhu 1 ( eu eu ) and Coshu 1 ( eu eu ) 2 2 Sinhu eu e u 2. tanhu Coshu eu e u and Cothu Sinhu eu eu Coshu eu eu 1 2 1 2 3. Sechu and Cschu Coshu eu e u Sinhu eu e u 4. Cosh2u Sinh2u 1 5. tanh2 u Sech2u 1 and Coth2u Csch2u 1 Coshu Sinhu e u and Cosh( u ) Coshu and Cosh0 1 and Sinh0 0 Sinh( x y ) Sinhx .Coshy Coshx.Sinhy Cosh( x y ) Coshx .Coshy Sinhx.Sinhy Sinh2 x 2.Sinhx .Coshx Cosh2 x Cosh 2 x Sinh 2 x Coshu Sinhu e u Sinh( u ) Sinhu Cosh 2 x Cosh2 x 1 Sinh 2 x Cosh2 x 1 and 2 2 y=Sinhx y=Cschx y=Cschx ١٨
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y Sinhx Dx : x and Ry : y y Coshx Ry : y 1 y tanh x
y=Sechx y=Cothx 1 y=tanhx -1 y=Cothx y Sinhx Dx : x and Ry : y y Coshx Ry : y 1 y tanh x Ry : 1 y 1 y Cothx Dx : x 0 Ry : y 1 or y 1 y Sechx Ry : 0 y 1 y Cschx Ry : y 0 EX-16- Let tanh u = - 7 / 25 , determine the values of the remaining five hyperbolic functions . Sol.- ١٩
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49 eln x e ln x e ln x e ln x e ln x ln x x 2 1 x 2 1
1 25 Cothu tanh u 7 49 tanh 2 u Sech2 u 1 Sech2 u 1 Sechu 24 625 25 1 25 Sechu 24 Coshu tanh u Sinhu 7 Sinhu Sinhu 7 Coshu 25 25 24 24 1 24 Cschu Sinhu 7 EX-17- Rewrite the following expressions in terms of exponentials . Write the final result as simply as you can : a ) 2Cosh(ln x ) c ) Cosh5 x Sinh5 x b ) tanh(ln x ) d ) ( Sinhx Coshx )4 Sol.- eln x e ln x 1 x a ) 2Cosh(ln x ) 2. x 2 e ln x e ln x b ) tanh(ln x ) e e ln x ln x x 1 x 2 x 1 x 1 x 2 1 x e5 x e 5 x e5 x e 5 x c ) Cosh5 x Sinh5 x e5 x 2 2 e x e x e x e x 4 d ) ( Sinhx Coshx )4 e 4 x 2 2 EX-18- Solve the equation for x : Cosh x = Sinh x + 1 / 2 . Sol. - Coshx Sinhx 1 e x 1 x ln 1 ln 2 x ln 2 2 2 EX-19 – Verify the following identity : Sinh(u+v)=Sinh u. Cosh v + Cosh u.Sinh v then verify Sinh(u-v)=Sinh u. Cosh v - Cosh u.Sinh v Sol.- ٢٠
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