asind

Asind is the inverse sine function when the output is given in degrees. It returns, for a given value x, the angle θ in degrees such that sin θ = x, with θ constrained to the range of arcsin in degrees, namely −90° to 90°.

Definition and relation to radians: For x in the interval [−1, 1], asind(x) = θ where θ ∈ [−90°, 90°] and sin θ = x. It is related to the standard inverse sine in radians, asin, by asind(x) = (180/π) · asin(x). Consequently, asind(x) = 180/π · arcsin(x) in radians.

Domain, range, and properties: The domain of asind is [−1, 1], and its range is [−90°, 90°]. The function is odd (asind(−x) = −asind(x)), continuous and strictly increasing on its domain. Its derivative is d/dx asind(x) = 180/(π · √(1 − x²)) for |x| < 1, with appropriate handling at the endpoints.

Computing and usage: In programming languages and calculators that offer deg-mode inverse sine functions, asind is commonly provided as a built-in function (e.g., in MATLAB and Octave). In environments that use radians, asind can be computed as deg(asin(x)) or by multiplying the radian result by 180/π. Examples: asind(0.5) = 30, asind(1) = 90, asind(−0.7) ≈ −44.427°.

Notes: For x outside [−1, 1], the real-valued asind is undefined and typically yields NaN or an error; complex-valued extensions exist in some mathematical contexts.