2.3. INVERSE TRIGONOMETRIC FUNCTIONS 31 Arccot x, are multi-valued. The following relations define the multi-valued inverse trigono- metric functions: sin  Arcsin x  = x,cos  Arccos x  = x, tan  Arctan x  = x,cot  Arccot x  = x. These functions admit the following verbal definitions: Arcsin x is the angle whose sine is equal to x; Arccos x is the angle whose cosine is equal to x;Arctanx is the angle whose tangent is equal to x; Arccot x is the angle whose cotangent is equal to x. The principal (single-valued) branches ofthe inverse trigonometric functions are denoted by arcsin x ≡ sin –1 x (arcsine is the inverse of sine), arccos x ≡ cos –1 x (arccosine is the inverse of cosine), arctan x ≡ tan –1 x (arctangent is the inverse of tangent), arccot x ≡ cot –1 x (arccotangent is the inverse of cotangent) and are determined by the inequalities – π 2 ≤ arcsin x ≤ π 2 , 0 ≤ arccos x ≤ π (–1 ≤ x ≤ 1); – π 2 <arctanx < π 2 , 0 < arccot x < π (–∞ < x < ∞). The following equivalent relations can be taken as definitions of single-valued inverse trigonometric functions: y =arcsinx,–1 ≤ x ≤ 1 ⇐⇒ x =siny,– π 2 ≤ y ≤ π 2 ; y = arccos x,–1 ≤ x ≤ 1 ⇐⇒ x =cosy, 0 ≤ y ≤ π; y =arctanx,–∞ < x <+∞⇐⇒x =tany,– π 2 < y < π 2 ; y = arccot x,–∞ < x <+∞⇐⇒x =coty, 0 < y < π. The multi-valued and the single-valued inverse trigonometric functions are related by the formulas Arcsin x =(–1) n arcsin x + πn, Arccos x = arccos x + 2πn, Arctan x =arctanx + πn, Arccot x = arccot x + πn, where n = 0, 1, 2, The graphs of inverse trigonometric functions are obtained from the graphs of the corresponding trigonometric functions by mirror reflection with respect to the straight line y = x (with the domain of each function being taken into account). 2.3.1-2. Arcsine: y =arcsinx. This function is defined for all x [–1, 1] and its range is y [– π 2 , π 2 ]. The arcsine is an odd, nonperiodic, bounded function that crosses the axes Ox and Oy at the origin x = 0, y = 0. This is an increasing function in its domain, and it takes its smallest value y =– π 2 at the point x =–1; it takes its largest value y = π 2 at the point x = 1. The graph of the function y =arcsinx is given in Fig. 2.10. 32 ELEMENTARY FUNCTIONS 2.3.1-3. Arccosine: y = arccos x. This function is defined for all x [–1, 1], and its range is y [0, π]. It is neither odd nor even. It is a nonperiodic, bounded function that crosses the axis Oy at the point y = π 2 and crosses the axis Ox at the point x = 1. This is a decreasing function in its domain, and at the point x =–1 it takes its largest value y = π; at the point x = 1 it takes its smallest value y = 0.Forallx in its domain, the following relation holds: arccos x = π 2 –arcsinx.The graph of the function y = arccos x is given in Fig. 2.11. O 1 yxarcsin= x y 1 π 2 π 2 Figure 2.10. The graph of the function y =arcsinx. O 1 π x y yx= arccos 1 π 2 Figure 2.11. The graph of the function y = arccos x. 2.3.1-4. Arctangent: y =arctanx. This function is defined for all x, and its range is y (– π 2 , π 2 ). The arctangent is an odd, nonperiodic, bounded function that crosses the coordinate axes at the origin x = 0, y = 0. This is an increasing function on the real axis with no points of extremum. It has two horizontal asymptotes: y =– π 2 (as x → –∞)andy = π 2 (as x → +∞). The graph of the function y =arctanx is given in Fig. 2.12. 2.3.1-5. Arccotangent: y = arccot x. This function is defined for all x, and its range is y (0, π). The arccotangent is neither odd nor even. It is a nonperiodic, bounded function that crosses the axis Oy at the point y = π 2 and does not cross the axis Ox. This is a decreasing function on the entire real axis with no points of extremum. It has two horizontal asymptotes y = 0 (as x → +∞)andy = π (as x →–∞). For all x, the following relation holds: arccot x = π 2 –arctanx. The graph of the function y = arccot x is given in Fig. 2.13. O 1 x y 1 yxarctan= π 2 π 2 Figure 2.12. The graph of the function y =arctanx. O 1 x y π 1 yxarccot= π 2 Figure 2.13. The graph of the function y = arccot x. 2.3. INVERSE TRIGONOMETRIC FUNCTIONS 33 2.3.2. Properties of Inverse Trigonometric Functions 2.3.2-1. Simplest formulas. sin(arcsin x)=x, cos(arccos x)=x, tan(arctan x)=x, cot(arccot x)=x. 2.3.2-2. Some properties. arcsin(–x) = – arcsin x, arctan(–x) = – arctan x, arccos(–x)=π – arccos x, arccot(–x)=π – arccot x, arcsin(sin x)=  x – 2nπ if 2nπ – π 2 ≤ x ≤ 2nπ + π 2 , –x + 2(n + 1)π if (2n + 1)π – π 2 ≤ x ≤ 2(n + 1)π + π 2 , arccos(cos x)=  x – 2nπ if 2nπ ≤ x ≤ (2n + 1)π, –x + 2(n + 1)π if (2n + 1)π ≤ x ≤ 2(n + 1)π, arctan(tan x)=x – nπ if nπ – π 2 < x < nπ + π 2 , arccot(cot x)=x – nπ if nπ < x <(n + 1)π. 2.3.2-3. Relations between inverse trigonometric functions. arcsin x+arccos x = π 2 ,arctanx+arccot x = π 2 ; arcsin x = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ arccos √ 1 –x 2 if 0 ≤ x ≤ 1, – arccos √ 1 –x 2 if –1 ≤ x ≤ 0, arctan x √ 1 –x 2 if –1 < x < 1, arccot √ 1 –x 2 x –π if –1 ≤ x < 0; arccos x = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ arcsin √ 1 –x 2 if 0 ≤ x ≤ 1, π –arcsin √ 1 –x 2 if –1 ≤ x ≤ 0, arctan √ 1 –x 2 x if 0 < x ≤ 1, arccot x √ 1 –x 2 if –1 < x < 1; arctan x = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ arcsin x √ 1 +x 2 for any x, arccos 1 √ 1 +x 2 if x ≥ 0, – arccos 1 √ 1 +x 2 if x ≤ 0, arccot 1 x if x > 0; arccotx = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ arcsin 1 √ 1 +x 2 if x > 0, π –arcsin 1 √ 1 +x 2 if x < 0, arctan 1 x if x > 0, π +arctan 1 x if x < 0. 2.3.2-4. Addition and subtraction of inverse trigonometric functions. arcsin x +arcsiny =arcsin  x  1 – y 2 + y √ 1 – x 2  for x 2 + y 2 ≤ 1, arccos x arccos y = arccos  xy  (1 – x 2 )(1 – y 2 )  for x y ≥ 0, 34 ELEMENTARY FUNCTIONS arctan x +arctany =arctan x + y 1 – xy for xy < 1, arctan x –arctany =arctan x – y 1 + xy for xy >–1. 2.3.2-5. Differentiation formulas. d dx arcsin x = 1 √ 1 – x 2 , d dx arccos x =– 1 √ 1 – x 2 , d dx arctan x = 1 1 + x 2 , d dx arccot x =– 1 1 + x 2 . 2.3.2-6. Integration formulas.  arcsin xdx= x arcsin x + √ 1 – x 2 + C,  arccos xdx= x arccosx – √ 1 – x 2 + C,  arctan xdx= x arctanx – 1 2 ln(1 + x 2 )+C,  arccot xdx= x arccotx + 1 2 ln(1 + x 2 )+C, where C is an arbitrary constant. 2.3.2-7. Expansion in power series. arcsin x = x + 1 2 x 3 3 + 1×3 2×4 x 5 5 + 1×3×5 2×4×6 x 7 7 + ···+ 1×3×···× (2n – 1) 2×4×···× (2n) x 2n+1 2n + 1 + ··· (|x| < 1), arctan x = x – x 3 3 + x 5 5 – x 7 7 + ···+(–1) n–1 x 2n–1 2n – 1 + ··· (|x| ≤ 1), arctan x = π 2 – 1 x + 1 3x 3 – 1 5x 5 + ···+(–1) n 1 (2n – 1)x 2n–1 + ··· (|x| > 1). The expansions for arccos x and arccot x can be obtained from the relations arccos x = π 2 –arcsinx and arccot x = π 2 –arctanx. 2.4. Hyperbolic Functions 2.4.1. Definitions. Graphs of Hyperbolic Functions 2.4.1-1. Definitions of hyperbolic functions. Hyperbolic functions are defi ned in terms of the exponential functions as follows: sinh x = e x – e –x 2 ,coshx = e x + e –x 2 ,tanhx = e x – e –x e x + e –x ,cothx = e x + e –x e x – e –x . The graphs of hyperbolic functions are given below. 2.4. HYPERBOLIC FUNCTIONS 35 2.4.1-2. Hyperbolic sine: y =sinhx. This function is defined for all x and its range is the entire y-axis. The hyperbolic sine is an odd, nonperiodic, unbounded function that crosses the axes Ox and Oy at the origin x = 0, y = 0. This is an increasing function in its domain with no points of extremum. The graph of the function y =sinhx is given in Fig. 2.14. 2.4.1-3. Hyperbolic cosine: y =coshx. This function is defined for all x, and its range is y [1,+∞). The hyperbolic cosine is a nonperiodic, unbounded function that crosses the axis Oy at the point 1 and does not cross the axis Ox. This function is decreasing on the interval (–∞, 0) and is increasing on the interval (0,+∞); it takes its smallest value y = 1 at x = 0. The graph of the function y =coshx is given in Fig. 2.15. O 1 1 2 x y 1 1 2 yxsinh= Figure 2.14. The graph of the function y =sinhx. O 1 1 2 2 3 4 x y 1 2 yxcosh= Figure 2.15. The graph of the function y =coshx. 2.4.1-4. Hyperbolic tangent: y =tanhx. This function is defi ned for all x, and its range is y (–1, 1). The hyperbolic tangent is an odd, nonperiodic, bounded function that crosses the coordinate axes at the origin x = 0, y = 0. This is an increasing function on the entire real axis and has two horizontal asymptotes: y =–1 (as x → –∞)andy = 1 (as x → +∞). The graph of the function y =tanhx is given in Fig. 2.16. 2.4.1-5. Hyperbolic cotangent: y =cothx. This function is defined for all x ≠0, and its range consists of all y (–∞,–1)andy (1,+∞). The hyperbolic cotangent is an odd, nonperiodic, unbounded function that does not cross the coordinate axes. This is a decreasing function on each of the semiaxes of its domain; it has no points of extremum and does not cross the coordinate axes. It has two horizontal asymptotes: y =–1 (as x → –∞)andy = 1 (as x → +∞). The graph of the function y =cothx is given in Fig. 2.17. 36 ELEMENTARY FUNCTIONS O 1 1 2 x y 1 1 2 yxtanh= Figure 2.16. The graph of the function y =tanhx. O 1 x y 1 yxcoth= Figure 2.17. The graph of the function y =cothx. 2.4.2. Properties of Hyperbolic Functions 2.4.2-1. Simplest relations. cosh 2 x –sinh 2 x = 1, sinh(–x)=–sinhx, tanh x = sinh x cosh x , tanh(–x)=–tanhx, 1 –tanh 2 x = 1 cosh 2 x , tanh x coth x = 1, cosh(–x)=coshx, coth x = cosh x sinh x , coth(–x)=–cothx, coth 2 x – 1 = 1 sinh 2 x . 2.4.2-2. Relations between hyperbolic functions of single argument (x ≥ 0). sinh x =  cosh 2 x – 1 = tanh x √ 1 –tanh 2 x = 1 √ coth 2 x – 1 , cosh x =  sinh 2 x + 1 = 1 √ 1 –tanh 2 x = coth x √ coth 2 x – 1 , tanh x = sinh x √ sinh 2 x + 1 = √ cosh 2 x – 1 cosh x = 1 coth x , coth x = √ sinh 2 x + 1 sinh x = cosh x √ cosh 2 x – 1 = 1 tanh x . 2.4.2-3. Addition formulas. sinh(x y)=sinhx cosh y sinh y cosh x,cosh(x y)=coshx cosh y sinh x sinh y, tanh(x y)= tanh x tanh y 1 tanh x tanh y ,coth(x y)= coth x coth y 1 coth y coth x . 2.4. HYPERBOLIC FUNCTIONS 37 2.4.2-4. Addition and subtraction of hyperbolic functions. sinh x sinh y = 2 sinh  x y 2  cosh  x y 2  , cosh x +coshy = 2 cosh  x + y 2  cosh  x – y 2  , cosh x –coshy = 2 sinh  x + y 2  sinh  x – y 2  , sinh 2 x –sinh 2 y =cosh 2 x –cosh 2 y =sinh(x + y)sinh(x – y), sinh 2 x +cosh 2 y =cosh(x + y)cosh(x – y), (cosh x sinh x) n =cosh(nx) sinh(nx), tanh x tanh y = sinh(x y) cosh x cosh y ,cothx coth y = sinh(x y) sinh x sinh y , where n = 0, 1, 2, 2.4.2-5. Products of hyperbolic functions. sinh x sinh y = 1 2 [cosh(x + y)–cosh(x – y)], cosh x cosh y = 1 2 [cosh(x + y)+cosh(x – y)], sinh x cosh y = 1 2 [sinh(x + y)+sinh(x – y)]. 2.4.2-6. Powers of hyperbolic functions. cosh 2 x = 1 2 cosh 2x+ 1 2 , cosh 3 x = 1 4 cosh 3x+ 3 4 cosh x, cosh 4 x = 1 8 cosh 4x+ 1 2 cosh 2x+ 3 8 , cosh 5 x = 1 16 cosh 5x+ 5 16 cosh 3x+ 5 8 cosh x, sinh 2 x = 1 2 cosh 2x– 1 2 , sinh 3 x = 1 4 sinh 3x– 3 4 sinh x, sinh 4 x = 1 8 cosh 4x– 1 2 cosh 2x+ 3 8 , sinh 5 x = 1 16 sinh 5x– 5 16 sinh 3x+ 5 8 sinh x, cosh 2n x = 1 2 2n–1 n–1  k=0 C k 2n cosh[2(n –k)x]+ 1 2 2n C n 2n , cosh 2n+1 x = 1 2 2n n  k=0 C k 2n+1 cosh[(2n –2k +1)x], sinh 2n x = 1 2 2n–1 n–1  k=0 (–1) k C k 2n cosh[2(n –k)x]+ (–1) n 2 2n C n 2n , sinh 2n+1 x = 1 2 2n n  k=0 (–1) k C k 2n+1 sinh[(2n –2k +1)x]. Here, n = 1, 2, ;and C k m are binomial coefficients. . each of the semiaxes of its domain; it has no points of extremum and does not cross the coordinate axes. It has two horizontal asymptotes: y =–1 (as x → –∞)andy = 1 (as x → +∞). The graph of the. with no points of extremum. It has two horizontal asymptotes y = 0 (as x → +∞)andy = π (as x →–∞). For all x, the following relation holds: arccot x = π 2 –arctanx. The graph of the function. Addition and subtraction of inverse trigonometric functions. arcsin x +arcsiny =arcsin  x  1 – y 2 + y √ 1 – x 2  for x 2 + y 2 ≤ 1, arccos x arccos y = arccos  xy  (1 – x 2 )(1 – y 2 )  for