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)
In radians this is sin-1(½)
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)
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
inverse trig functions, interval notation