Inverse Cosecant
csc^{1} 
cosec^{1} 
Csc^{1} 
Cosec^{1} 
arccsc 
arccosec 
Arccsc 
Arccosec 
The inverse function of cosecant.
Basic idea: In order to find csc^{1} 2,
we ask "what
angle has cosecant equal to 2?" The
answer is 30°.
As a result we say csc^{1} 2
= 30°.
In radians this is csc^{1} 2
= π/6.
More: There are actually many angles that have cosecant equal
to 2. We are really asking "what is the simplest, most basic
angle that has cosecant equal to 2?" As before, the answer
is 30°. Thus csc^{1} 2
= 30° or csc^{1} 2 = π/6.
Details: What is csc^{1} (–2)?
Do we choose 210°, –30°,
330°, or some other angle? The answer
is –30°.
With inverse cosecant, we select the angle on the right half of
the unit circle having measure
as close to zero as possible. Thus csc^{1} (–2)
= –30° or csc^{1} (–2) = –π/6.
In
other words, the range of csc^{1} is
restricted to [–90°, 0) U (0,
90°] or .
Note: csc 0 is undefined, so 0 is not in the range of csc^{1}.
Note: arccsc refers to "arc cosecant",
or the radian measure of the arc on a circle corresponding to
a given value of cosecant.
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. Csc or Csc^{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
