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The
graph of the tangent function
and the
cotangent function |
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The
graph of the
arc-tangent function and the arc-cotangent function |
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| Trigonometric
(cyclometric) functions and inverse trigonometric functions (arc
functions) |
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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). |
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| sin2x
+ cos2x
= 1 |
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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. |
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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. |
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| The
graph of the tangent function
and the
cotangent function |
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- 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, |
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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
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, |
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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. |
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| The
graph of the
arc-tangent function and the
arc-cotangent function |
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