The inverse function of sine.
**Basic idea**: To find sin^{-1}(½), we ask "what angle has sine equal to
½?" The answer is 30°. As a result we
say sin^{-1}(½) = 30°. In radians this is
sin^{-1}(½)
= π/6.
**More**: There are actually many
angles that have sine equal to ½. We are
really asking "what is the simplest, most basic
angle that has sine equal to ½?" As before,
the answer is 30°. Thus sin^{-1}(½)
= 30° or sin^{-1}(½)
= π/6.
**Details**: What is sin^{-1}(-½)? Do we choose 210°, -30°,
330° , or some other angle? The answer is
-30°. With inverse sine, we select the
angle on the right half of the unit circle having
measure as close to zero as possible. Thus sin^{-1}(-½)
= -30° or sin^{-1}(-½)
= -π/6.
In other words, the range of sin^{-1}
is restricted to [-90°, 90°]
or .
Note: arcsin refers to "arc sine", or the radian
measure of the arc on a circle corresponding
to a given value of sine.
__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.
Sin or Sin^{-1}).
However, most mathematicians do not follow this
practice. This website does not distinguish
between capitalized and uncapitalized trig functions.
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