**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 ^{-1}(½) = 30°.^{-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 ^{-1}(½) = 30°^{-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 ^{-1}(–½) = –30°^{–1}(–½) = –π/6.

In other words, the range of ^{-1}

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 functionssine, 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 ^{-1}).

**See also**

Inverse trigonometry, inverse trig functions, interval notation

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