Lecture 5: Inverse Trigonometric Functions 5 The inverse sine function The function f(x = sin(x is not one-to-one on (,, but is on [ π, π Moreover, f still has range [, when restricte to this interval Hence it is reasonable to restrict f to [ π, π to obtain an inverse for the sine function Definition We call the inverse of the sine function, when restricte to [ π, π, the arcsine function, with value at a point x enote by arcsin(x or sin (x In other wors, y = sin (x if an only if sin(y = x an π y π Note that the arcsine function has omain [, an range [ π, π sin (0 = 0, sin ( = π, an sin ( = π, ( sin = π 6 Note that sin(sin (x = x for any x an sin (sin(x = x for any π x However, the latter equality oes not hol for other values of x For example, π sin (sin ( ( 7π = sin = π 6 6 Since for x in [, we have sin(sin (x = x, it follows that x sin(sin (x = x x Hence cos(sin (x x sin (x =, 5-
an so Lecture 5: Inverse Trigonometric Functions 5- x sin (x = Now if y = sin (x, then π y π Since cos(y 0 for π y π, we have Thus we have the following proposition cos(sin (x an sin(y = x Thus cos (y = sin (y = x cos(sin (x = cos(y = x Proposition x sin (x = x for all < x < x sin (3x = 6x 9x Note that f(x = sin (x is an increasing function on [,, with graph as shown below 5 5 05 05-5 - -05 05 5-05 -5 - -05 05 5-05 - - -5-5 Graphs of y = sin(x an y = sin (x We now have the integration formula x = x sin (x + c
Lecture 5: Inverse Trigonometric Functions 5-3 Using the substitution u = x, we have 0 x = x 0 u u = sin (u 0 = ( sin sin (0 = π 5 The inverse tangent function Note that the function f(x = tan(x is one-to-one if its omain is restricte to ( π, π Moreover, f still has range (, when restricte to this interval Hence it is reasonable to restrict f to ( π, π to obtain an inverse for the tangent function Definition We call the inverse of the tangent function, when restricte to ( π, π, the arctangent function, with value at a point x enote by arctan(x or tan (x In other wors, y = tan (x if an only if tan(y = x an π < y < π Note that the arctangent function has omain (, an range ( π, π an tan (0 = 0, tan ( = π, tan ( 3 = π 3 Since tan(tan (x = x for all x in (,, we have x tan(tan (x = x x Hence sec (tan (x x tan (x =,
an so Lecture 5: Inverse Trigonometric Functions 5- x tan (x = Now if y = tan (x, then π < y < π Thus we have the following proposition sec (tan (x an tan(y = x Thus sec (y = + tan (y = + x Proposition x tan (x = + x If f(x = tan ( x, then f (x = ( ( x + = 8 + x Note that f(x = tan (x is an increasing function on (, with an Hence the lines y = π an y = π lim x tan (x = π lim x tan (x = π are horizontal asymptotes for the graph of f Moreover, f (x = ( + x x (x = ( + x, from which it follows that the graph of f is concave upwar on (, 0 an concave ownwar on (0, The graph is shown below 5 05-5 - -05 05 5 - - -05 - - - -5 Graphs of y = tan(x an y = tan (x
Lecture 5: Inverse Trigonometric Functions 5-5 We now have the integration formula + x x = tan (x + c + x x = Using the substitution u = x, we have + ( x x = + u u = tan (u + c = ( x tan + c 53 The inverse secant function Note that the function f(x = sec(x is one-to-one if its omain is restricte to [ 0, π [ π, 3π Moreover, f still has range (, [, when restricte to this omain Hence it is reasonable to restrict f to this omain to obtain an inverse for the secant function Definition We call the inverse of the secant function, when restricte to [ 0, π [ π, 3π, the arcsecant function, with value at a point x enote by arcsec(x or sec (x In other wors, y = sec (x if an only if sec(y = x an either 0 y < π or π y < 3π Note that the arcsecant function has omain (, [, an range [ 0, π [ π, 3π sec ( = 0, sec ( = π,
Lecture 5: Inverse Trigonometric Functions 5-6 an sec ( = π 3 Since sec(sec (x = x, we have Hence an so Now if y = sec (x, then x sec(sec (x = x x sec(sec (x tan(sec (x x sec (x =, x sec (x = x tan(sec (x tan (y = sec (y = x Since tan(y 0 for y in the range of the arcsecant function, we have tan(sec (x = x Thus we have the following proposition Proposition For all x or x, x sec (x = x x The previous proposition gives us the integration formula x x x = sec (x + c The graph of f(x = sec (x is shown below Notice how the vertical asymptotes of g(x = sec(x are transforme into horizontal asymptotes for f 0 75 5 5-5 -5-75 -0 3 3-0 -5 5 0 Graphs of y = sec(x an y = sec (x
Lecture 5: Inverse Trigonometric Functions 5-7 5 The remaining inverse trigonometric functions The remaining inverse trigonometric functions are not use as frequently as the three we have consiere above They are efine as follows: y = cos (x if an only if cos(y = x an 0 y π, y = cot (x if an only if cot(y = x an 0 < y < π, an y = csc (x if an only if csc(y = x an 0 < y π or π < y 3π One may show that an x cos (x =, x x cot (x = + x, x csc (x = x x Note that these are just the negations of the erivatives of the inverses of the corresponing co-functions In particular, they o not give us any new integration formulas Moreover, this implies that these functions iffer by only a constant from the inverses of their corresponing co-functions