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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 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.

 

 

 

See also

Inverse trigonometry, inverse trig functions, interval notation

 


  this page updated 27-aug-12
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes