Differences between Inverse and Reciprocal Trig Functions

Inverse The inverse of a trigonometric function is the function that "undoes" the function. Commonly these are denoted with the prefix "arc-". So, the inverse of the tan function is the arctan function. By "undoing" the function, we mean, for instance, that: arctan(tan x) = x = tan(arctan x) arcsin(sin x) = x = sin(arcsin x) arccos(cos x) = x = cos(arccos x)

Reciprocal The reciprocal of a trigonometric function is simply its algebraic reciprocal--that is, 1 over the function. For instance, the reciprocal of sin x is 1 / sin x These have identities as other trigonometric functions. For instance, 1 / sin x ≡ csc x, 1 / cos x ≡ sec x, 1 / tan x ≡ cot x

To summarize Inverse Used to find the value of an angle. tan ⁻ ¹ x ≡ arctan x The inverse of a trigonometric function gives you the angle instead of the ratio. Reciprocal (tan x) ⁻ ¹ ≡ 1 / tan x The reciprocal of a trigonometric function is simply its algebraic reciprocal--that is, 1 over the function. For instance, the reciprocal of sin x is 1 / sin x Cosecant csc Secant sec Cotangent Cot