Final answer:
The arctan or arctangent function, with the symbol tan⁻¹, serves to find the angle whose tangent is a given real number and has a range between (-π/2, π/2). It is widely used in trigonometry for determining angles in right triangles and for vector direction calculations.
Step-by-step explanation:
The function tan⁻¹, also known as arctangent or arctan, is the inverse of the tangent function with domain ℝ (all real numbers) and range (-π/2, π/2). Its main purpose is to determine the angle whose tangent is a given number. For instance, if we have the ratio of the opposite side to the adjacent side in a right triangle, we can use arctan to find the angle associated with that ratio.
The arctan function is crucial in trigonometry and has applications in various fields such as physics, engineering, and computer science. It is directly related to trigonometric identities such as sin (a ± β), cos (α ± β), and tan (a ± β), although it performs a different function by being the inverse of the tangent. Just like subtraction is the inverse of addition, and division is the inverse of multiplication, arctan is the inverse function of the tangent.