2 HYPERBOLIC FUNCTIONS


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1 HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions; be able to fin inverse hyperbolic functions an use them in calculus applications; recognise logarithmic equivalents of inverse hyperbolic functions.. Introuction This chapter will introuce you to the hyperbolic functions which you may have notice on your calculator with the abbreviation hyp. You will see some connections with trigonometric functions an will be able to fin various integrals which cannot be foun without the help of hyperbolic functions. The first systematic consieration of hyperbolic functions was one by the Swiss mathematician Johann Heinrich Lambert (78777).. Definitions The hyperbolic cosine function, written cosh x, is efine for all real values of x by the relation cosh x = ( ex + e x ) Similarly the hyperbolic sine function, sinh x, is efine by sinh x = ( ex e x ) The names of these two hyperbolic functions suggest that they have similar properties to the trigonometric functions an some of these will be investigate. 33
2 Chapter Hyperbolic Functions Activity Show that cosh x + sinh x = e x an simplify cosh x sinh x. an By multiplying the expressions for cosh x + sinh x ( cosh x sinh x) together, show that (b) By consiering cosh x sinh x = ( cosh x + sinh x ) + ( cosh x sinh x) show that cosh x + sinh x = cosh x (c) By consiering ( cosh x + sinh x ) ( cosh x sinh x) show that sinh x cosh x = sinh x Activity Use the efinitions of sinh x an cosh x in terms of exponential functions to prove that cosh x = cosh x (b) cosh x = + sinh x Example Prove that cosh( x y)=cosh x cosh y sinh x sinh y Solution 34 cosh x cosh y = ( ex + e x ) ( ey +e y ) = 4 ex + y + e x y + e ( x y) +e ( x+y) sinh x sinh y = ( ex e x ) ( ey e y ) Subtracting gives = 4 ex + y e x y e ( x y) +e ( x+y) cosh x cosh y sinh x sinh y = ( 4 ex y + e ( x y) ) = ex y + e x y =cosh x y
3 Chapter Hyperbolic Functions Exercise A Prove the following ientities.. sinh( x) = sinh x (b) cosh( x) = cosh x. sinh( x + y) = sinh x cosh y + cosh x sinh y (b) sinh( x y) = sinh x cosh y cosh x sinh y 3. cosh( x + y) = cosh x cosh y + sinh x sinh y 4. sinh A + sinh B = sinh A + B cosh 5. cosh A cosh B = sinh A + B sinh A B A B. Osborn's rule You shoul have notice from the previous exercise a similarity between the corresponing ientities for trigonometric functions. In fact, trigonometric formulae can be converte into formulae for hyperbolic functions using Osborn's rule, which states that cos shoul be converte into cosh an sin into sinh, except when there is a prouct of two sines, when a sign change must be effecte. For example, cosx = sin x can be converte, remembering that sin x = sin x.sin x, into cosh x = + sinh x. But sin A = sin Acos A simply converts to sinh A = sinh A cosh A because there is no prouct of sines. Activity 3 Given the following trigonometric formulae, use Osborn's rule to write own the corresponing hyperbolic function formulae. sin A sin B = cos A + B sin A B (b) sin 3A = 3sin A 4sin 3 A (c) cos θ + sin θ =.3 Further functions Corresponing to the trigonometric functions tan x, cot x, sec x an cosecx we efine tanh x = sinh x cosh x, coth x = tanh x = cosh x sinh x, 35
4 Chapter Hyperbolic Functions sechx = cosh x an cosechx = sinh x By implication when using Osborn's rule, where the function tanh x occurs, it must be regare as involving sinh x. Therefore, to convert the formula sec x = + tan x we must write sech x = tanh x. Activity 4 Prove that tanh x = ex e x e x + e an sechx = x e x + e, x an hence verify that sech x = tanh x. (b) Apply Osborn's rule to obtain a formula which correspons to cosec y = + cot y. Prove the result by converting cosechy an coth y into exponential functions..4 Graphs of hyperbolic functions You coul plot the graphs of cosh x an sinh x quite easily on a graphics calculator an obtain graphs as shown opposite. y cosh x x The shape of the graph of y = cosh x is that of a particular chain supporte at each en an hanging freely. It is often calle a catenary (from the Latin wor catena for chain or threa). y sinh x x 36
5 Chapter Hyperbolic Functions Activity 5 (b) (c) Superimpose the graphs of y = cosh x an y = sinh x on the screen of a graphics calculator. Do the curves ever intersect? Use a graphics calculator to sketch the function f : x a tanh x with omain x R. What is the range of the function? Try to preict what the graphs of y = sechx, y=cosechx an y = coth x will look like. Check your ieas by plotting the graphs on a graphics calculator..5 Solving equations Suppose sinh x = 3 4 an we wish to fin the exact value of x. Recall that cosh x = + sinh x an cosh x is always positive, so when sinh x = 3 4, cosh x = 5 4. From Activity, we have sinh x + cosh x = e x so e x = = an hence x = ln. Alternatively, we can write sinh x = ( ex e x ) so sinh x = 3 4 means ( ex e x )= 3 4 e x 3 e x = an multiplying by e x e x 3e x = ( e x ) ( e x +)= e x = or e x = But e x is always positive so e x = x = ln. 37
6 Chapter Hyperbolic Functions Activity 6 Fin the values of x for which cosh x = 3 5 expressing your answers as natural logarithms. Example Solve the equation cosh x +sinh x = 5 giving your answer in terms of a natural logarithm. Solution cosh x = ( e x + e x ); sinh x = ( e x e x ) So e x + e x + 5e x 5e x = 5 6e x 5 4e x = 6e 4 x 5e x 4 = ( 3e x 4) ( e x +)= e x = 4 3 or e x = The only real solution occurs when e x > So x = ln 4 3 x = ln 4 3 Exercise B. Given that sinh x = 5, fin the values of cosh x (b) tanh x (c) sechx () coth x (e) sinh x (f) cosh x Determine the value of x as a natural logarithm.. Given that cosh x = 5, etermine the values of 4 sinh x (b) cosh x (c) sinh x Use the formula for cosh x + x to etermine the value of cosh3x. 38
7 Chapter Hyperbolic Functions 3. In the case when tanh x =, show that x = ln3. 4. Solve the following equations giving your answers in terms of natural logarithms. 4cosh x + sinh x = 4 (b) 3sinh x cosh x = (c) 4tanh x = + sechx 5. Fin the possible values of sinh x for which cosh x + 7sinh x = 4 (You may fin the ientity cosh x sinh x = useful.) Hence fin the possible values of x, leaving your answers as natural logarithms. 6. Solve the equations 3cosh x + 5cosh x = (b) 4cosh x sinh x = 7 7. Express 5cosh x 4sinh x in the form Rcosh x α giving the values of R an tanhα. Hence write own the minimum value of 5cosh x 4sinh x an fin the value of x at which this occurs, giving your answer in terms of a natural logarithm. 8. Determine a conition on A an B for which the equation Acosh x + Bsinh x = has at least one real solution. 9. Given that a, b, c are all positive, show that when a>b then acosh x + bsinh x can be written in the form Rcosh( x + α ). Hence etermine a further conition for which the equation acosh x + bsinh x = c has real solutions.. Use an appropriate iterative metho to fin the solution of the equation cosh x = 3x giving your answer correct to three significant figures..6 Calculus of hyperbolic functions Activity 7 (b) (c) By writing cosh x = ( ex + e x ), prove that ( cosh x )=sinh x. Use a similar metho to fin ( sinh x). Assuming the erivatives of sinh x an cosh x, use the quotient rule to prove that if y = tanh x = sinh x cosh x then y =sech x. Note: care must be taken that Osborn's rule is not use to obtain corresponing results from trigonometry in calculus. 39
8 Chapter Hyperbolic Functions Activity 8 Use the quotient rule, or otherwise, to prove that (b) (c) ( sechx )= sechx tanh x ( cosechx )= cosechx coth x cothx = cosech x Example Integrate each of the following with respect to x. cosh 3x (b) sinh x (c) x sinh x () e x cosh x Solution cosh 3x= sinh 3x + constant 3 (b) sinh x can be foun by using cosh x = + sinh x giving ( cosh x ) = 4 sinh x x + constant Alternatively, you coul change to exponentials, giving sinh x = ( 4 e x + e x ) sinh x= e x 8 x 8 e x +constant Can you show this answer is ientical to the one foun earlier? (c) Using integration by parts, x sinh x= xcosh x cosh x = x cosh x sinh x + constant 4
9 Chapter Hyperbolic Functions () Certainly this is foun most easily by converting to exponentials, giving e x cosh x = e x + e x cosh x= 4 ex + x+constant Exercise C. Differentiate with respect to x tanh 4x (b) sechx (c) cosech( 5x +3) () sinh e x (e) cosh 3 x (f ) tanh( sin x) (g) cosh 5x sinh 3x (h) ( coth 4x). Integrate each of the following with respect to x. sinh 4x (b) cosh 3x (c) x cosh x () sech 7x (e) cosechx coth3x (f ) tanh x (g) tanh x (h) e sinh3x (i) x cosh( x 3 + 4) ( j) sinh 4 x (k) cosh xsinh3x (l) sechx 4. A curve has equation y = λ cosh x + sinh x where λ is a constant. Sketch the curve for the cases λ = an λ =. (b) Determine the coorinates of the turning point of the curve in the case when λ = 4 3. Is this a maximum or minimum point? (c) Determine the range of values of λ for which the curve has no real turning points. 5. Fin the area of the region boune by the coorinate axes, the line x = ln3 an the curve with equation y = cosh x + sech x. 3. Fin the equation of the tangent to the curve with equation y = 3cosh x sinh x at the point where x = ln..7 Inverse hyperbolic functions The function f : x a sinh x ( x R) is oneone, as can be seen from the graph in Section.4. This means that the inverse function f exists. The inverse hyperbolic sine function is enote by sinh x. Its graph is obtaine by reflecting the graph of sinh x in the line y = x. Recall that ( sinh x )=cosh x, so the graient of the graph of y = sinh x is equal to at the origin. Similarly, the graph of sinh x has graient at the origin. y sinh x x 4
10 Chapter Hyperbolic Functions Similarly, the function g: x a tanh x ( x R)is oneone. You shoul have obtaine its graph in Activity 5. The range of g is y: < y <. { } or the open interval, y y = tanh x x Activity 9 Sketch the graph of the inverse tanh function, tanh x. Its range is now R. What is its omain? y y = cosh x The function h: x a cosh x ( x R) is not a oneone function an so we cannot efine an inverse function. However, if we change the omain to give the function f : x a cosh x with omain { x: x R, x } then we o have a oneone function, as illustrate. So, provie we consier cosh x for x, we can efine the inverse function f : x a cosh x with omain { x: x R, x }. This is calle the principal value of cosh x. x y y = cosh x x.8 Logarithmic equivalents Activity Let y = sinh x so sinh y = x. Since cosh y is always positive, show that cosh y = ( + x ) By consiering sinh y + cosh y, fin an expression for e y in terms of x. Hence show that sinh x = ln x + + x 4
11 Chapter Hyperbolic Functions Activity Let y = tanh x so tanh y = x. Express tanh y in the terms of e y an hence show that e y = + x x. Deuce that tanh x = ln + x x. (Do not forget that tanh x is only efine for x <. ) Activity Let y = cosh x, where x, so cosh y = x. Use the graph in Section.7 to explain why y is positive an hence why sinh y is positive. Show that sinh y = ( x ). Hence show that cosh x = ln x + ( x ). The full results are summarise below. sinh x = ln x + x + tanh x = ln + x x (all values of x) ( x < ) cosh x = ln x + ( x ) ( x ) 43
12 Chapter Hyperbolic Functions Exercise D. Express each of the following in logarithmic form. sinh 3 (b) cosh (c) tanh 4. Given that y = sinh x, fin a quaratic equation satisfie by e y. Hence obtain the logarithmic form of sinh x. Explain why you iscar one of the solutions. 3. Express sech x in logarithmic form for < x <. 4. Fin the value of x for which sinh x + cosh ( x + ) =. 5. Solve the equation tanh x = ln. x +.9 Derivatives of inverse hyperbolic functions Let y = sinh x so that x = sinh y. y =cosh y but cosh y = sinh y + = x + an cosh y is always positive. So y = ( x +) an therefore y = x + In other wors, ( sinh x)= x + Activity 3 Show that ( cosh x)= x (b) Show that ( tanh x)= x An alternative way of showing that ( sinh x)= to use the logarithmic equivalents. x + is 44
13 Chapter Hyperbolic Functions Since [ ] =+.x ( x +) x+ x + =+ x x + = ( x +) + x x + we can now fin the erivative of ln[ x + ( x +) ] { } ln x + x + which cancels own to x + + x = x + x +. x+ x + So ( sinh x)= x + You can use a similar approach to fin the erivatives of cosh x an tanh x but the algebra is a little messy. Example Differentiate cosh x + (b) sinh x with respect to x ( x > ). Solution Use the function of a function or chain rule. [ cosh ( x +) ]=. {( x +) } = 4x +4x = ( x + x) (b) sinh x = x. x + = x. x +x = x + x 45
14 Chapter Hyperbolic Functions Exercise E Differentiate each of the expressions in Questions to 6 with respect to x.. cosh 4 + 3x. sinh x 3. tanh 3x + 4. x sinh ( x) 5. cosh x x > 6. sinh ( coshx) 7. Differentiate sech x with respect to x, by first writing x = sechy. 8. Fin an expression for the erivative of cosech x in terms of x. 9. Prove that ( coth x) = ( x ).. Use of hyperbolic functions in integration Activity 4 Use the results from Section.9 to write own the values of an (b) x + ( x ) Activity 5 Differentiate sinh x 3 with respect to x. Hence fin. ( x + 9) What o you think is equal to? ( x + 49) Activity 6 Use the substitution x = cosh u to show that =cosh x x 4 + constant 46
15 Chapter Hyperbolic Functions Activity 7 Prove, by using suitable substitutions that, where a is a constant, (b) =sinh x x + a a + constant =cosh x x a a + constant Integrals of this type are foun by means of a substitution involving hyperbolic functions. They may be a little more complicate than the ones above an it is sometimes necessary to complete the square. Activity 8 Express 4x 8x 5 in the form A( x B) +C, where A, B an C are constants. Example Evaluate in terms of natural logarithms 7 4 4x 8x 5 Solution From Activity 8, the integral can be written as 7 4 { 4( x ) 9} You nee to make use of the ientity cosh A = sinh A because of the appearance of the enominator. Substitute 4( x ) =9 cosh u in orer to accomplish this. So ( x )=3cosh u an u = 3sinh x 47
16 Chapter Hyperbolic Functions The enominator then becomes { 9cosh u 9} = ( 9sinh u) =3sinh u In orer to eal with the limits, note that when ( [ ]) an when ( [ ]) x = 4, cosh u = so u = ln + 3 x = 7, cosh u = 4 so u = ln The integral then becomes cosh 4 3 sinh uu = 3sinh u cosh cosh 4 cosh u { } = [ cosh 4 cosh ]= ln ( + 3 ) ln Example Evaluate 3 ( x + 6x +3) leaving your answer in terms of natural logarithms. Solution Completing the square, x + 6x +3 = ( x + 3) +4 You will nee the ientity sinh A + = cosh A this time because of the + sign after completing the square. Now make the substitution x + 3 = sinh θ giving θ = cosh θ When x = 3, sinh θ = θ = When x =, sinh θ = θ = sinh = ln( + 5) The integral transforms to sinh ( 4sinh θ + 4). cosh θ θ sinh = 4 cosh θ θ 48
17 Chapter Hyperbolic Functions You can either convert this into exponentials or you can use the ientity cosh A = cosh A giving sinh + cosh θ θ sinh = [ θ + sinhθ] sinh = [ θ + sinhθ coshθ] = ( sinh + + ) = ln( + 5)+4 5 Activity 9 Show that an euce that sec 3 φ φ = ( sec φ tan φ φ )=sec3 φ sec φ Hence use the substitution { sec φ tan φ + ln ( sec φ + tan φ )} x + 3 = tan φ in the integral of the previous example an verify that you obtain the same answer. 49
18 Chapter Hyperbolic Functions Exercise F Evaluate the integrals in Questions to... x x + x x 4 x + 6x x 6. x 9 7. x ( + x + ) x +. 4 x 9 x + x x + x.. 5 ( 3x x + 8 ) x 4 + 4x Use the substitution u = e x to evaluate sech x. 4. Differentiate sinh x with respect to x. By writing sinh x as sinh x, use integration by parts to fin sinh. (b) Use integration by parts to evaluate 5. Evaluate each of the following integrals. 3 (b) 3 x + 6x 7 x + 6x 7 6. Transform the integral x 4 + x by means of the substitution x =. Hence, by u means of a further substitution, or otherwise, evaluate the integral. 7. The point P has coorinates ( acosht, bsinht). Show that P lies on the hyperbola with equation x a y b =. Which branch oes it lie on when a>? Given that O is the origin, an A is the point (a, ), prove that the region boune by the lines OA an OP an the arc AP of the hyperbola has area abt. cosh x.. Miscellaneous Exercises. Miscellaneous Exercises. The sketch below shows the curve with equation y = 3cosh x xsinh x, which cuts the yaxis at the point A. Prove that, at A, y takes a minimum value an state this value. y A B y = 33cosh x xsinh x x Given that y = at B, show that the xcoorinate of B is the positive root of the equation x cosh x sinh x =. Show that this root lies between.8 an. Fin, by integration, the area of the finite region boune by the coorinate axes, the curve with equation y = 3cosh x xsinh x an the line x =, giving your answer in terms of e. (AEB) x 5
19 Chapter Hyperbolic Functions. Starting from the efinition coshθ = eθ + e θ show that an sinhθ = ( eθ e θ ) sinh(a + B) = sinh Acosh B + cosh Asinh B. There exist real numbers r an α such that 5cosh x +3sinh x rsinh( x + α ). Fin r an show that α = ln 3. Hence, or otherwise, solve the equation 5cosh x +3sinh x = sinh (b) show that = ln 5e 5cosh x +3sinh x 3e + 3. Given that x <, prove that = tanh x Show by integrating the result above that tanh x = ln + x x Use integration by parts to fin tanh x. 4. Given that t tanh x, prove the ientities sinh x = t (b) cosh x = + t t t Hence, or otherwise, solve the equation 5. Solve the equations cosh ln x sinh x cosh x = tanh x sinh ln x = 3 4 (AEB) x. 7. Define cosechx an coth x in terms of exponential functions an from your efinitions prove that coth x + cosech x. Solve the equation 3coth x + 4cosechx = 3 8. Solve the equation 3sech x + 4 tanh x + = (AEB) 9. Fin the area of the region R boune by the curve with equation y = cosh x, the line x = ln an the coorinate axes. Fin also the volume obtaine when R is rotate completely about the xaxis. (AEB). Prove that sinh x = ln x + + x an write own a similar expression for cosh x. Given that cosh y 7sinh x = 3 an cosh y 3sinh x =, fin the real values of x an y in logarithmic form. (AEB). Evaluate the following integrals 3 x + 6x + 5 (b) x + 6x Use the efinitions in terms of exponential functions to prove that tanh x + tanh x = sechx (b) tanh x = tanh x Hence, use the substitution t = tanh x to fin (b) 4sinh x + 3e x + 3 = 6. Show that cosh x + sinh x = e x Deuce that cosh nx+sinh nx = Obtain a similar expression for cosh nx sinh nx. Hence prove that n k= n k coshn k xsinh k x cosh 7x = 64cosh 7 x cosh 5 x + 56cosh 3 x 7cosh x sechx (AEB) 3. Sketch the graph of the curve with equation y = tanh x an state the equations of its asymptotes. Use your sketch to show that the equation tanh x = 3x has just one root α. Show that α lies between 3 an3 3. Taking 3 as a first approximation for α, use the NewtonRaphson metho once to obtain a secon approximation, giving your answer to four ecimal places. 5
20 Chapter Hyperbolic Functions 4. Prove that sinh x = ln x + + x. Given that exp( z) e z, show that satisfies the ifferential y = exp sinh x equation ( + x ) y y +x y= (b) Fin the value of sinh x, leaving your answer in terms of a natural logarithm. 5. Sketch the graph of y = tanh x. Determine the value of x, in terms of e, for which tanh x =. The point P is on the curve y = tanh x where y =. Fin the equation of the tangent to the curve at P. Determine where the tangent to the curve crosses the yaxis. 6. Evaluate the following integrals, giving your answers as multiples of π or in logarithmic form. 3x 6x+4 (b) 7. Fin the value of x for which +6x 3x sinh sinh x = sinh Starting from the efinitions of hyperbolic functions in terms of exponential functions, show that cosh( x y) = cosh x cosh y sinh xsinh y an that tanh = ln + x where < x <. x Fin the values of R an α such that 5cosh x 4sinh x Rcosh( x α ) Hence write own the coorinates of the minimum point on the curve with equation y = 5cosh x 4sinh x (b) Solve the equation 9sech y 3tanh y = 7, leaving your answer in terms of natural logarithms. (AEB) 9. Solve the equation 3sech x + 4tanh x + =. (AEB). Define sinh y an cosh y in terms of exponential functions an show that cosh y + sinh y y = ln cosh y sinh y By putting tanh y =, euce that 3 tanh 3 = ln (AEB). Show that the function f( x)= ( +x)sinh( 3x ) has a stationary value which occurs at the intersection of the curve y = tanh( 3x ) an the straight line y + 3x + 3 =. Show that the stationary value occurs between an. Use NewtonRaphson's metho once, using an initial value of to obtain an improve estimate of the xvalue of the stationary point.. Given that tanh x = sinh x, express the value cosh x of tanh x in terms of e x an e x. (b) Given that tanh y = t, show that y = ln + t t for < t <. (c) Given that y = tanh ( sin x) show that y =sec x an hence show that π 6 sec x tanh ( sin x)= 3 3. Solve the simultaneous equations 3 + ln3 (AEB) cosh x 3sinh y = sinh x + 6cosh y = 5 giving your answers in logarithmic form. (AEB) 4. Given that x = cosh y, show that the value of x is either ln x + Solve the equations ( x ) or ln x x sinh θ 5coshθ + 7 = (b) cosh( z + ln3) = ( ) 5
21 Chapter Hyperbolic Functions 5. Sketch the curve with equation y = sechx an etermine the coorinates of its point of inflection. The region boune by the curve, the coorinate axes an the line x = ln3 is R. Calculate the area of R (b) the volume generate when R is rotate through π raians about the xaxis. 6. Solve the equation tanh x + 4sechx = Prove the ientity cosh x cos x sinh xsin x + coshx cosx 8. Prove that 6sinh x cosh 3 x cosh5x + cosh3x cosh x Hence, or otherwise, evaluate 6sinh x cosh 3 x, giving your answer in terms of e. 9. Show that the minimum value of sinh x + ncosh x is ( n ) an that this occurs when x = ln n n + Show also, by obtaining a quaratic equation in e x, that if k > ( n ) then the equation sinh x + ncosh x = k has two real roots, giving your answers in terms of natural logarithms. (AEB) 53
22 Chapter Hyperbolic Functions 54
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