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Chapter 3 65

# Chapter 3 65 - SECTION 3.9 HYPERBOLIC FUNCTIONS 3.9...

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Unformatted text preview: SECTION 3.9 HYPERBOLIC FUNCTIONS 3.9 Hyperbolic Functions 1. 12. 13. 14. 15. 16. . coshac + sinhx 2 l(e\$ + 8β0β) + 10. 11. (a) sinhO = Β§(eΒ° 2 eβ) = 0 (b) coshO = Β§(e0 +e0) = 5(1 + 1) = 1 0 β0 1 _ β1 2 _ 1 (e βe W βeββe:e 20.76159 (a) tanh02m20 (b)ta.nh1β β¬1+β¬_1 82-1-1 β _ 1 . 61,)2_e_1n2 elu2_(eln2) 1 2β2 1_2_Β§ βΒ§ (a) s1nh(ln 2) _ 2 _ 2 _ 2 _ 2 _ 4 (b) sinh2 = gong? 2 6-2) z 3.62686 ln3 βln3 1 6 +6 3+ - 5 (a) cosh3 = Β§(e3 + 673) z 10.06766 (b) cosh(ln 3) = f = 2 3 : g (a) sechO 2 coslhO 2 % 2 1 (b) cosh_1 1 2 0 because coshO 2 1. (a) sinhl = %(61 β eβ1)% 117520 (b) Using Equation 3, we have sinhβl 1 =1n(1+ 1/12 +1) :1n(1 + V?) 2 0.88137. sinh(βac) 2 %[eβm β e_(_1)] 2 ο¬eβ - em) 2 β%(eββ 2 6β3β) 2 βsinh:z: c0sh(βx) 2 %[e_z + e_(β\$)] 2 ο¬eβ + em) 2 ο¬eaβ + 51ββ) 2 coshm 1 1 2 Β§( 5 cosha: β sinha: 2 %(eββ + eβ) β ο¬ex β 6β3β) 2 %(2eβw) 2 6β1 sinhmcoshy + coshmsinhy 2 [% (e:C β 5β1)][%( y 2i[(e\$:ylezy 635+?! ezy)l(eβy ex y+6 rigβ31yβ 2 %(2e\$+y β 26β00β31) 2 Hemο¬β β {WW} 2 sinh(m + y) coshxcoshy + sinhmsinhy 2 [% (em ββ 64)] [% (ey + 6βy)] + [% (ex β CβEH [1(ey 2 679)] : iibβiο¬y + 8117β + 8ββ1β ββ 6β95β11) + (61er _ ew-y _ e~z+y + e-mβy)i I ο¬2ez+y + 26β70β11) : Heme: + eβ(z+y)] : c0sh(a: + y) Divide both sides of the identity cosh2 m β sinh2 a: 2 1 by sinh2 cc: cosh2 \$ sinh2 :1: 1 sinh2 ac sinh2 an sinh2 m (I) (:0ch 9B _ 1 : CSCh2 23β sinh :1: cosh y cosh m sinh y tanh(a: + y) 2 w ~ sinhazcoshy + coshmsinh y 2 W cosh(m + y) cosh ac cosh y + sinh m sinh y COSh 3: COSh y Slnh :1: Slnh y cosh a: cosh y cosh x cosh y ~ tanh :L' + tanh y _ 1+ tanh J) tanh 11 Putting y 2 a: in the result from Exercise 1 1, we have sinh 2x 2 sinh(m + x) 2 sinh x cosh x + cosh m sinh a: 2 2 sinh a; cosh 2:. Putting y 2 x in the result from Exercise 12, we have cosh 2m 2 cosh(a: + 3:) 2 cosh ac cosh a: + sinh ac sinh cc 2 cosh2 m + sinh2 ac. Z15 ...
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