CHAPTER 24 HYPERBOLIC FUNCTIONS
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1 EXERCISE 00 Pag 5 CHAPTER HYPERBOLIC FUNCTIONS. Evaluat corrct to significant figurs: (a) sh 0.6 (b) sh , corrct to significant figurs (a) sh ( ) Altrnativly, using a scintific calculator, using hyp, sin , corrct to significant figurs (b) sh ( ) Altrnativly, using a scintific calculator, using hyp, sin Evaluat corrct to significant figurs: (a) ch 0.7 (b) ch.65.7, corrct to significant figurs (a) ch ( ) Altrnativly, using a scintific calculator, using hyp, cos , corrct to significant figurs (b) ch ( ) Altrnativly, using a scintific calculator, using hyp, cos Evaluat corrct to significant figurs: (a) th 0.65 (b) th.8 (a) th , corrct to significant figurs Altrnativly, using a scintific calculator, using hyp, tan (b) Using a calculator, tanh , corrct to significant figurs. Evaluat corrct to significant figurs: (a) cosch 0.5 (b) cosch. 70 0, John Bird
2 (a) cosch 0.5 sh , corrct to significant figurs, using a calculator (b) cosch. sh , corrct to significant figurs, using a calculator 5. Evaluat corrct to significant figurs: (a) sch 0.9 (b) sch.67 (a) sch 0.9 ch , corrct to significant figurs, using a calculator (b) sch.67 ch , corrct to significant figurs, using a calculator 6. Evaluat corrct to significant figurs: (a) coth 0. (b) coth.8 (a) coth 0. (b) coth.8 th 0. th.8.98, corrct to significant figurs, using a calculator.05, corrct to significant figurs, using a calculator 7. A tlgraph wir hangs so that its shap is dscribd by y 50 ch 50. Evaluat, corrct to significant figurs, th valu of y whn 5 Whn 0.5, y y 50ch ch ch , corrct to significant figurs 8. Th lngth l of a havy cabl hanging undr gravity is givn by l c sh(l/c). Find th valu of l whn c 0 and L 0 0 l csh( L/ c ) (0) sh 80sh 0.7 (0) 8 7 0, John Bird
3 9. V 0.55L tanh(6.d/l) is a formula for vlocity V of wavs ovr th bottom of shallow watr, whr d is th dpth and L is th wavlngth. If d 8.0 and L 96, calculat th valu of V. V 0.55Ltanh(6. d / L) (6.)(8.0) 0.55(96) tanh tanh Hnc, V , John Bird
4 EXERCISE 0 Pag 9. Prov th idntitis: (a) ch(p Q) ch P ch Q sh P sh Q (b) ch ch sh (a) R.H.S. ch P ch Q sh P sh Q P P Q Q P P Q Q P Q P Q P Q P Q P Q P Q P Q P Q PQ PQ PQ ( PQ) cosh(p Q) L.H.S. (b) R.H.S. ch sh cosh L.H.S.. Prov th idntitis: (a) coth cosch th (b) ch θ sh θ (a) R.H.S. cosch th sh sh sh sh sh ch sh ch ch sh ch ch sh ch ch ch coth L.H.S sh ch sh θ θ 7 0, John Bird (b) R.H.S. sh θ ( θ θ )( θ θ) [ θ θ θ θ θ θ]
5 θ θ [ θ 0 0 θ ] [ θ θ] θ θ ch θ L.H.S.. Prov th idntitis: (a) th(a B) th A th B th Ath B (b) sh A sh A ch A (a) R.H.S. AA BB th A th B A A B B th th A A B B A B A A B B ( A A)( B B) ( A A)( BB) ( A A)( B B) ( A A)( B B) ( AA)( BB) ( A A)( B B) ( A A)( B B) ( A A)( B B) ( A A)( BB) ( A A)( B B) ( AA)( BB) (AB ABABA B) (AB AB ABA B) (AB AB AB A B) (ABAB AB A B) A B A B A B A B A B A B ( AB) ( AB) A B A B ( AB) ( AB) ( ( AB) ( AB) ) sinh( A B) (( AB) ( AB) ) cosh( A B) tanh(a B) L.H.S. (b) R.H.S. sh A ch A A A A A A A A A A A A A sinh A L.H.S.. Prov th idntitis: (a) sh(a B) sh A ch B ch A sh B (b) 7 sh ch tanh ch coth 0, John Bird
6 (a) R.H.S. sh A ch B ch A sh B A A B B A A B B A B A B A B A B A B A B A B A B [ ] ( A B) ( A B ) [ A B ] sh( A B) L.H.S ( A B) (b) L.H.S. sh ch sh sh ch ch coth ch sh sinc ch sh sh sh sh tanh ch ch ch R.H.S. 5. Givn P Q 6 ch sh, find P and Q. 6 ( ) ( ) P Q 6ch sh i.. P Q from which, P and Q 6. If 5 A sh B ch, find A and B. A A B B 5 Ash B ch A B A B A B A B AB i.. 5 A B Hnc, 5 i.. A B 0 () 75 0, John Bird
7 and A B i.. A B 8 () () () givs: A 8 from which, A 9 From () B 76 0, John Bird
8 EXERCISE 0 Pag. Solv, corrct to dcimal placs: (a) sinh (b) sh A. (a) If sinh thn sinh 0.88 This can b dtrmind by calculator. (i) Prss hyp (ii) Choos, which is sinh (iii) Typ in (iv) Clos brackt ) (v) Prss and th answr is (b) If sinh A. thn sinh (.).609 by calculator. Solv, corrct to dcimal placs: (a) cosh B.87 (b) ch (a) If cosh B.87, thn B cosh.87.8 This can b dtrmind by calculator. (i) Prss hyp (ii) Choos 5, which is cosh (iii) Typ in.87 (iv) Clos brackt ) (v) Prss and th answr is.80 With rfrnc to Figur., pag 6, it can b sn that thr will b two valus corrsponding to cosh.87 Hnc, B ±.8 (b) If cosh, thn cosh significant figurs and cosh cosh , corrct to With rfrnc to Figur., pag 6, it can b sn that thr will b two valus corrsponding to cosh.5. Hnc, ±0.96. Solv, corrct to dcimal placs: (a) tanh y 0.76 (b) th. (a) This can b dtrmind by calculator (i) Prss hyp (ii) Choos 6, which is tanh (iii) Typ in 0.76 (iv) Clos brackt ) (v) Prss and th answr is i.. th solution of tanh y 0.76 is: 0.996, corrct to dcimal placs 77 0, John Bird
9 (b) If tanh., thn tanh. significant figurs. and tanh tanh , corrct to. Solv, corrct to dcimal placs: (a) sch B 0.5 (b) sch Z (a) If sch B 0.5, thn B sch 0.5 cosh 0.5 sinc cosh sch and using a calculator, B.7, corrct to dcimal placs With rfrnc to th graph of y sch in Figur., pag 7, it can b sn that thr will b two valus corrsponding to sch B 0. Hnc, B ±.7 (b) If sch Z 0.889, thn sch cosh sinc cosh sch and using a calculator, 0.97, corrct to dcimal placs With rfrnc to th graph of y sch in Figur., pag 7, it can b sn that thr will b two valus corrsponding to sch Z Hnc, Z ± Solv, corrct to dcimal placs: (a) cosch θ.5 (b) 5 cosch.5 (a) If cosch θ.5, thn θ cosch (.5) sinh.5 sinc sinh cosch i.. θ 0.6, corrct to dcimal placs.5 5 (b) If 5 cosch.5, thn cosch sinh 5.5 sinc sinh cosch i , corrct to dcimal placs 6. Solv, corrct to dcimal placs: (a) coth.5 (b) coth y.6 (a) If coth.5, thn coth.5 tanh.5 78 sinc tanh coth 0, John Bird
10 i.. 0.6, corrct to dcimal placs.6 coth coth.8 tanh.8 (b) If coth y.6, thn y ( ) sinc tanh coth i.. y 0.676, corrct to dcimal placs 7. Solv corrct to dcimal placs:.5 sh.5 ch 0.5 sh.5 ch 0 i i and or 0.5 i i.. Hnc, ln 0.5 from which, 0.5 ln Solv, corrct to dcimal placs: sh ch 5 sh ch 5 i.. 5 i and or Multiplying ach trm by givs:.5( ) i...5( ) , John Bird
11 Using th quadratic formula: ± ± (.5) 5 5 ( 5) (.5)(0.5) 5 0 i or from which, i.. ln(.897 ) or ln( ) or.8, corrct to dcimal placs 9. Solv corrct to dcimal placs: th 0 th 0 i.. and 0 i.. ( ) Hnc, 5 0 and 5 Thus, 5 5 from which, i.. ln 5 and 5 ln A chain hangs so that its shap is of th form y 56 cosh. Dtrmin, corrct to 56 significant figurs, (a) th valu of y whn is 5, and (b) th valu of whn y is 6.5 (a) Whn 5, 5 y 56ch ( / 56) 56ch , using a calculator (b) Whn, y 6.5, thn ch ( / 56) Thus, 6.5 ch or i Thus, i , John Bird
12 from which, (.679) (.679 ) ( )( ) ( ) ±.679 ± or Hnc, ln.609 or ln i.. 56 ln or 56 ln Hnc, ± , John Bird
13 EXERCISE 0 Pag. Us th sris pansion for ch to valuat, corrct to dcimal placs: (a) ch.5 (b) ch 0.8 (a) ch! ch.5... Lt.5, thn! i.. ch.5.5, corrct to dcimal placs, which may b chckd by using a calculator (b) ch! ch Lt 0.8, thn! i.. ch 0.8.7, corrct to dcimal placs, which may b chckd by using a calculator. Us th sris pansion for sh to valuat, corrct to dcimal placs: (a) sh 0.5 (b) sh (a) sh 5.. Lt, thn! 5! sh ! 5! 7! i.. sh , corrct to dcimal placs (b) sh 5.. Lt, thn! 5! sh ! 5! 7! 9!! , John Bird
14 i.. sh.669, corrct to dcimal placs. Epand th following as a powr sris as far as th trm in 5 : (a) sh (b) ch (a) sh ( ) ( ) ( ) ! 5! 6 5 ( ) ( ) 9 8 as far as th trm in (b) ch......!! as far as th trm in. Prov th idntity: sh θ sh θ θ 7 6 θ 0 θ 5 L.H.S. ( ) ( ) θ θ 5 θ θ5 sh θ shθ θ... θ...! 5!! 5! 8 θ θ θ... θ θ θ θ θ θ θ θ θ θ R.H.S. 6 0 ( ) 5 5 as far as th trm in θ 5 only θ θ θ θ θ θ5 5. Prov th idntity: sh ch θ L.H.S. ( θ /) ( θ /) ( θ /) ( θ /) 5 θ θ θ sh ch......! 5!!! 8 (0) 8 (6)() θ θ θ5 θ θ 5 θ θ θ θ5 θ as far as th trm in θ 5 only , John Bird
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