Inverse
Cosine
cos^{1}
Cos^{1}
arccos
Arccos
The
inverse function of cosine.
Basic idea: To find cos^{1} (½),
we ask "what
angle has cosine equal to ½?" The
answer is 60°. As a result we say cos^{1} (½)
= 60°.
In radians this is cos^{1} (½)
= π/3.
More: There are actually many angles that have cosine equal to ½.
We are really asking "what is the simplest, most basic angle that has
cosine equal to ½?" As before, the
answer is 60°. Thus cos^{1} (½)
= 60° or cos^{1} (½) = π/3.
Details: What is cos^{1} (–½)?
Do we choose 120°, –120°, 240°,
or some other angle?
The answer is 120°.
With inverse cosine, we select the angle on the top half of the unit
circle. Thus cos^{1} (–½)
= 120° or
cos^{1} (–½) = 2π/3.
In
other words, the range of cos^{1} is
restricted to [0, 180°] or [0, π].
Note: arccos refers to "arc cosine",
or the radian measure of the arc on a circle corresponding to
a given value of cosine.
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. Cos or Cos^{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
