SSCE1693 ENGINEERING MATHEMATICS CHAPTER 1: FURTHER TRANSCENDENTAL FUNCTIONS WAN RUKAIDA BT WAN ABDULLAH YUDARIAH BT MOHAMMAD YUSOF SHAZIRAWATI BT MOHD PUZI NUR ARINA BAZILAH BT AZIZ ZUHAILA BT ISMAIL Department of Mathematical Sciences Facult of Sciences Universiti Teknologi Malasia
1.1 Hperbolic Functions 1.1.1 Definition of Hperbolic Functions 1.1. Graphs of Hperbolic Functions 1.1.3 Hperbolic Identities 1. Inverse Functions 1..1 Inverse Trigonometric Functions 1.. Inverse Trigonometric Identities 1..3 Inverse Hperbolic Functions 1..4 Log Form of the Inverse Hperbolic Functions
1.1 Hperbolic Functions 1.1.1 Definition of Hperbolic Functions Hperbolic Sine, pronounced shine. x x e e sinh x = Hperbolic Cosine, pronounced cosh. x x e + e cosh x = Hperbolic Tangent, pronounced tanh. tanh x = x x x sinh x e e e 1 = cosh x x x e + e x e + 1 Hperbolic Secant, pronounced shek. 1 sech x = = x x cosh x e + e Hperbolic Cosecant, pronounced coshek. 1 cosech x = = x x sinh x e e Hperbolic Cotangent, pronounced coth. x x cosh x e + e coth x = = x x sinh x e e
1.1. Graphs of Hperbolic Functions Since the hperbolic functions depend on the values of e x and e x, its graphs is a combination of the exponential graphs. (i) Graph of sinh x From the graph, we see (a) sinh 0 = 0. (b) The domain is all real numbers. (c) The curve is smmetrical about the origin, i.e. sinh ( x) = sinh x (d) It is an increasing one-to-one function.
(ii) Graph of cosh x We see from the graph of = cosh x that: (a) cosh 0 = 1 (b) The domain is all real numbers. (c) The value of cosh x is never less than 1. (d) The curve is smmetrical about the -axis, i.e. cosh ( x) = cosh x (e) For an given value of cosh x, there are two values of x.
(iii) Graph of tanh x We see (a) tanh 0 = 0 (b) tanh x alwas lies between = 1 and = 1. (c) tanh ( x) = tanh x (d) It has horizontal asmptotes = ± 1.
1.1.3 Hperbolic Identities For ever identit obeed b trigonometric functions, there is a corresponding identit obeed b hperbolic functions. 1. cosh xx sinh xx = 1. 1 tanh xx = sech xx 3. coth xx = cosech xx 4. sinh(xx ± ) = sinh x cosh ± cosh x sinh 5. cosh(xx ± ) = cosh x cosh ± sinh x sinh 6. tanh(xx ± ) = tanh x ±tanh 1 ±tanh xx tanh 7. sinh xx = sinh xx cosh xx 8. cosh xx = cosh xx + sinh xx = cosh xx = sinh xx + 1 9. tanh xx = tanh xx 1+tanh xx
Some of the hperbolic identities follow exactl the trig. identities; others have a difference in sign. Trig. Identities Hperbolic Identities sech θ = 1 coshθ cosech θ = 1 sinhθ coth θ = 1 tanhθ cosh θ sinhθ 1 1 tanh θ sech θ cothθ cosech θ sinha cosha sinh Acosh A cosh 1+ sinh cosh A+ sinh A A 1 A
Examples 1.1 1. B using definition of hperbolic functions, (a) Evaluate sinh(-4) to four decimal places. (b) Show that cosh xx = cosh xx. B using identities of hperbolic functions, show that 1 tanh xx 1 + tanh xx = sech xx 3. Solve the following for x, giving our answer in 4dcp. cosh xx = sinh xx + 1 4. Solve for x if given cosh x sinh x =. 5. B using definition of hperbolic functions, proof that cosh xx sinh xx = 1 6. Solve cosh x = 4 sinh x. Use 4 dcp.
1. INVERSE FUNCTIONS Definition 1. (Inverse Functions) If f : X Y is a one-to-one function with the domain X and the range Y, then there exists an inverse function, f : where the domain is Y and the range is X such that Y X = f ( x) x = f ( ) Thus, 1 f ( f ( x)) = x for all values of x in the domain f. Note: The graph of inverse function is reflections about the line = x.
1..1 Inverse Trigonometric Functions Trigonometric functions are periodic hence the are not one-to one. However, if we restrict the domain to a chosen interval, then the restricted function is one-to-one and invertible. (i) Inverse Sine Function Look at the graph of = sin x shown below The function f ( x) = sin x is not one to one. But if the domain is restricted to one to one.,, then f(x) is
Definition: The inverse sine function is defined as where = sin x x = sin and 1 x 1. The function 1 sin x is sometimes written as arcsin x. The graph of = sin 1 x is shown below 4-1 1 4 ( ) ( ) 1 f x sin x = f x = arcsin x
(ii) Inverse Cosine Function Look at the graph of = cos x shown below The function f ( x) = cos x is not one to one. But if the domain is restricted to [ 0, ], then f(x) is one to one. Definition: The inverse cosine function is defined as = cos x x = cos where 0 and 1 x 1.
The graph of = cos 1 x is shown below -1 1 ( ) ( ) 1 f x cos x = f x = arccos x (iii) Inverse Tangent Function Look at the graph of = tan x shown below
The function f ( x) = tan x is not one to one. But if the domain is restricted to,, then f(x) is one to one. Definition: The inverse tangent function is defined as = tan 1 x x = tan where and x. The graph of = tan 1 x is shown below -1 1 ( ) ( ) = 1 f x tan x f x = arctan x
(iv) Inverse Cotangent Function Domain: Range: (v) Inverse Secant Function Domain: Range:
(vi) Inverse Cosecant Function Domain: Range:
Table of Inverse Trigonometric Functions Functions Domain Range = sin x [ 1, 1], = cos x [ 1, 1] [0, ] = tan x (, ), = csc x 1 = sec x 1,0 0, x 0,, x = cot 1 x (, ) (0, ) It is easier to remember the restrictions on the domain and range if ou do so in terms of quadrants. 1 sin x whereas sin x 1 (sin x) =. sin x
1.. Inverse Trigonometric Identities The definition of the inverse functions ields several formulas. Inversion formulas sin (sin 1 for 1 x 1 x ) = x sin (sin ) = for tan (tan 1 x ) = x for all x tan (tan ) = < < These formulas are valid onl on the specified domain Basic Relation sin tan x + cos cot x = x + x = for 0 x 1 for 0 x 1
sec x + csc 1 x = for 0 x 1 1
Negative Argument Formulas sin tan cos ( x) = sin x sec ( x) = sec x ( x) = tan x csc ( x) = csc x ( x) = cos x cot ( x) = cot x Reciprocal Identities 1 x csc x = sin 1 for x 1 1 x sec x = cos 1 for x 1 cot 1 xx = tan 1 1 xx for x 1
Examples 1.: 1. Evaluate the given functions. (i) sin (sin 0.5) (ii) sin (sin 1 ) (iv) sin (sin ) (iii) sin 1 (sin 0.5). Evaluate the given functions. (i) arcsec( ) (ii) csc ( ) (iii) cot 1 1 3 3. For 1 x 1, show that 1 1 (i) sin ( x) = sin x (ii) cos(sin x) = 1 x
1..3 Inverse Hperbolic Functions The three basic inverse hperbolic functions are 1 1 sinh x, cosh x, and tanh x. Definition (Inverse Hperbolic Function) 1 = sinh 1 x x = sinh for all x and R = cosh 1 x x = cosh for x 1 and 0 = tanh 1 x x = tanh for 1 x 1, R Graphs of Inverse Hperbolic Functions (i) = sinh 1 x Domain: Range:
(ii) = cosh 1 x Domain: Range: (iii) = tanh x Domain: Range:
1..4 Log Form of the Inverse Hperbolic Functions It ma be shown that (a) cosh 1 xx = ln xx + xx (b) sinh 1 xx = ln xx + xx + 1 (c) tanh 1 xx = 1 (d) coth 1 xx = 1 ln 1+xx 1 xx ln xx+1 xx 1 (e) sech 1 xx = ln 1+ 1 xx xx (f) cosech 1 xx = ln 1 + 1+xx xx xx
Inverse Hperbolic Cosine (Proof) If we let = cosh 1 xx, then Hence, xx = cosh = ee + ee. On rearrangement, xx = ee + ee. (ee ) xxee + 1 = 0 Hence, (using formula bb± bb 4aaaa ) aa Since ee > 0, ee = xx ± 4xx 4 = xx ± xx ee = xx + xx Taking natural logarithms, = cosh 1 xx = ln(xx + xx )
Proof for ssssssss 1 xx = sinh 1 xx xx = sinh = ee ee xx = ee ee (multipl with ee ) xxee = ee ee xxxx = 0 ee = xx ± xx + 1 Since ee > 0, ee = xx + xx + 1 Taking natural logarithms, = sinh 1 xx = ln xx + xx + 1 In the same wa, we can find the expression for tanh 1 xx in logarithmic form.
Examples 1.3: Evaluate 1) ) 3) 1 sinh (0.5) 1 cosh (0.5) 1 tanh ( 0.6)