Transcendental functions
Exponential and logarithmic functions, inverse functions
Trigonometric (cyclometric) functions and inverse trigonometric functions (arc-functions)
Transcendental functions:
· Exponential and logarithmic functions are mutually inverse functions
- Exponential function  y = ex   <=>  x = ln y,   e = 2.718281828...the base of the natural logarithm,
exponential function is inverse of the natural logarithm function, so that  eln x = x.
- Logarithmic function  y = ln x = log e x    <=>   x = e y,    where x > 0
the natural logarithm function is inverse of the exponential function, so that  ln(ex) = x.
- Exponential function  y = ax   <=>    x = loga  y,   where a > 0 and  a is not 1

 exponential function with base a is inverse of the logarithmic function, so that
- Logarithmic function y = log a x    <=>    x = a y,   where a > 0a is not 1 and x > 0
the logarithmic function with base a is inverse of the exponential function, so that  loga (ax) = x.
·  Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
Trigonometric functions are defined as the ratios of the sides of a right triangle containing the angle equal to the argument of the function in radians.
Or more generally for real arguments, trigonometric functions are defined in terms of the coordinates of the terminal point Q of the arc (or angle) of the unit circle with the initial point at P(1, 0).
 sin2x + cos2x = 1
-  The sine function  y = sin x is the y-coordinate of the terminal point of the arc x of the unit circle. The graph of the sine function is the sine curve or sinusoid.
In a right-angled triangle the sine function is equal to the ratio of the length of the side opposite the given angle to the length of the hypotenuse.
-  The arc-sine function  y = sin-1x or  y = arcsin x is the inverse of the sine function, so that its value for any argument is an arc (angle) whose sine equals the given argument.
 That is,  y = sin-1x if and only if  x = sin y.  For example,
Thus, the arc-sine function is defined for arguments between -1 and 1, and its principal values are by convention taken to be those between -p/2 and p/2.
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-  The cosine function  y = cos x is the x-coordinate of the terminal point of the arc x of the unit circle. The graph of the cosine function is the cosine curve or cosinusoid.
In a right-angled triangle the cosine function is equal to the ratio of the length of the side adjacent the given angle to the length of the hypotenuse.
-  The arc-cosine function  y = cos-1x or  y = arccos x is the inverse of the cosine function, so that its value for any argument is an arc (angle) whose cosine equals the given argument.
 That is,  y = cos-1x if and only if  x = cos y.  For example,
Thus, the arc-cosine function is defined for arguments between -1 and 1, and its principal values are by convention taken to be those between 0 and p.
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-  The tangent function  y = tan x is the ratio of the y-coordinate to the x-coordinate of the terminal point of the arc x of the unit circle, or it is the ratio of the sine function to the cosine function.
In a right-angled triangle the tangent function is equal to the ratio of the length of the side opposite the given angle to that of the adjacent side.
-  The arc-tangent function  y = tan-1x or  y = arctan x is the inverse of the tangent function, so that its value for any argument is an arc (angle) whose tangent equals the given argument.
 That is,  y = tan-1x if and only if  x = tan y.  For example,
Thus, the arc-tangent function is defined for all real arguments, and its principal values are by convention taken to be those strictly between -p/2 and p/2.
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-  The cosecant function  y = csc x is the reciprocal of the sine function.
In a right-angled triangle the cosecant function is equal to the ratio of the length of the hypotenuse to that of the side opposite to the given angle.
-  The arc-cosecant function  y = csc-1x or  y = arccsc x is the inverse of the cosecant function, so that its value for any argument is an arc (angle) whose cosecant equals the given argument.
 That is,  y = csc-1x if and only if  x = csc y.  For example,
Thus, the arc-cosecant function is defined for arguments less than -1 or greater than 1, and its principal values are by convention taken to be those between -p/2 and p/2.
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-  The secant function  y = sec x is the reciprocal of the cosine function.
In a right-angled triangle the secant function is equal to the ratio of the length of the hypotenuse to that of the side adjacent to the given angle.
-  The arc-secant function  y = sec-1x or  y = arcsec x is the inverse of the secant function, so that its value for any argument is an arc (angle) whose secant equals the given argument.
 That is,  y = sec-1x if and only if  x = sec y.  For example,
Thus, the arc-secant function is defined for arguments less than -1 or greater than 1, and its principal values are by convention taken to be those between 0 and p.
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-  The cotangent function  y = cot x is the reciprocal of the tangent function, or it is the ratio of the cosine function to the sine function.
In a right-angled triangle the cotangent function is equal to the ratio of the length of the side adjacent to the given angle to that of the side opposite it.
-  The arc-cotangent function  y = cot-1x or  y = arccot x is the inverse of the cotangent function, so that its value for any argument is an arc (angle) whose cotangent equals the given argument.
 That is,  y = cot-1x if and only if  x = cot y.  For example,
Thus, the arc-cotangent function is defined for all real arguments, and its principal values are by convention taken to be those strictly between 0 and p.
Calculus contents A
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