|
|
|
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
> 0,
a 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. |
-
- - -
- - - |
- 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. |
-
- - -
- - - |
- 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. |
-
- - -
- - - |
- 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. |
-
- - -
- - - |
- 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. |
-
- - -
- - - |
- 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. |
|
|