Inverse
Cotangent
cot^{1} 
ctg^{1} 
Cot^{1} 
Ctg^{1} 
arccot 
arcctg 
Arccot 
Arcctg 
The inverse function of cotangent.
Basic idea: To find cot^{1} 1,
we ask "what
angle has cotangent equal to 1?" The answer is 45°.
As a result we say cot^{1} 1
= 45°.
In radians this is cot^{1} 1
= π/4.
More: There are actually many angles that have
cotangent equal to 1. We are really asking "what is the
simplest, most basic angle that has cotangent equal to 1?" As
before, the answer is 45°. Thus cot^{1} 1 = 45° or cot^{1} 1 = π/4.
Details: What is cot^{1} (–1)?
Do we choose 135°, –45°,
315°, or some other angle? The answer is 135°.
With inverse cotangent, we select the angle on the top half of
the unit circle. Thus cot^{1} (–1)
= 135° or
cot^{1} (–1) = 3π/4.
In
other words, the range of cot^{1} is
defined to be the angles on the upper half of the unit circle as
pictured below. The range of cot^{1} is
restricted to (0, 180°) or (0, π).
Note: arccot refers to "arc cotangent",
or the radian measure of the arc on a circle corresponding to
a given value of cotangent.
Technical note: Since none of the six trig
functions sine,
cosine, tangent, cosecant, secant, and cotangent are onetoone,
their inverses are not functions.
Each trig function can have its
domain restricted, however, in order to make its inverse a function.
Some mathematicians write these restricted trig functions and their
inverses with an initial capital letter (e.g. Cot or Cot^{1}).
However, most mathematicians do not follow this practice. This
website does not distinguish between capitalized and uncapitalized
trig functions.
See
also
Inverse
trigonometry, inverse
trig functions, interval notation
