What is is cosecant the inverse of sine?

Yes, the cosecant function (csc) is indeed the reciprocal (or multiplicative inverse) of the sine function (sin).

  • Definition: csc(x) = 1 / sin(x)

  • Domain and Range: Because cosecant is the reciprocal of sine, it is undefined wherever sine is equal to zero. This occurs at integer multiples of π (e.g., 0, π, 2π, -π, etc.). The range of sine is [-1, 1], so the range of cosecant is (-∞, -1] ∪ [1, ∞).

  • Relationship: Sine and cosecant are reciprocal functions. Therefore, if you know the value of sine at a particular angle, you can easily find the cosecant by taking its reciprocal, and vice versa.

  • Graphical Representation: The graph of cosecant has vertical asymptotes at the points where the sine function crosses the x-axis.

  • Trigonometric Identities: Cosecant appears in various trigonometric identities, often as a substitute for 1/sin(x).