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)
In radians this is cos-1 (½)
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.
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 one-to-one,
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, interval notation