Inverse Sine
sin^{1}^{
}Sin^{1}^{
}arcsin
Arcsin
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 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. Sin or Sin^{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
