

The graphs of the
elementary functions 
Trigonometric
(cyclometric) functions and inverse trigonometric functions (arc
functions) 
The
graphs of the trigonometric
functions and inverse trigonometric functions or arcfunctions 
The
graph of the tangent function
and the
cotangent function 
The
graph of the
arctangent function and the arccotangent function 





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). 



sin^{2}x
+ cos^{2}x
= 1 



 The
tangent function
y
= tan x
is the ratio of the ycoordinate to
the xcoordinate
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 rightangled 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
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 rightangled 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
graph of the tangent function
and the
cotangent function 




 The
arctangent function
y
= tan^{}^{1}x
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^{}^{1}x
if and only if x
= tan
y.
For
example, 


Thus, the arctangent
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
arccotangent function
y
= cot^{}^{1}x
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^{}^{1}x
if and only if x
= cot
y.
For
example, 


Thus, the arccotangent
function is defined for all real arguments, and its principal
values are by
convention taken to be those strictly between 0
and p. 

The
graph of the
arctangent function and the
arccotangent function 









Intermediate
algebra contents 



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