The derivative of the arctan function, d/dx arctan(x) = 1/(1 + x²), has a geometric interpretation related to the unit circle. If we consider a point P(x, y) on the unit circle, the slope of the line OP, where O is the origin, is equal to y/x. Now, if we let theta be the angle that OP makes with the positive x-axis, then y = sin(theta) and x = cos(theta), and so y/x = tan(theta). Thus, the slope of OP is tan(theta). The derivative of arctan(x) gives the rate of change of the angle theta with respect to x, which is the slope of the line tangent to the curve y = arctan(x) at any point (x, arctan(x)). This derivative is equal to the reciprocal of the square of the secant of theta, 1/(sec²(theta)), which simplifies to cos²(theta) due to the Pythagorean identity sin²(theta) + cos²(theta) = 1 and the identity 1/cos(theta) = sec(theta).

The range of inverse trigonometric functions is restricted to ensure that they are bijective, meaning they are both injective (no horizontal line intersects the graph more than once) and surjective (they cover the entire range). This restriction is crucial because, without it, the function would not have a unique output for each input, violating the definition of a function. For example, the sine function is periodic and takes all values from -1 to 1 infinitely many times as it repeats every 2pi. However, its inverse, arcsin, must give a unique angle for each sine value, so its range is restricted to [-pi/2, pi/2], where it is monotonically increasing and covers all values from -1 to 1 exactly once. This ensures that every possible output of the sine function has a unique inverse under the arcsin function. Similarly, the ranges of arccos and arctan are restricted to [0, pi] and (-pi/2, pi/2), respectively, ensuring their bijectiveness.

Yes, inverse trigonometric functions can be expressed as power series through a method known as Taylor series expansion. For instance, the Taylor series expansion for arctan(x) is derived by integrating the geometric series for 1/(1 + x²), which is 1 - x² + x⁴ - x⁶ + ... . The integral of this series, term by term, gives the Taylor series for arctan(x), which is x - x³/3 + x⁵/5 - x⁷/7 + ... . This representation is particularly useful in numerical analysis and approximating the values of the inverse trigonometric functions for certain inputs. It provides a means to calculate the function values to a desired degree of accuracy by including more terms from the series.

Inverse trigonometric functions, particularly arcsin, arccos, and arctan, find substantial applications in navigation, especially in determining the direction or path along which a vehicle or an individual should move to reach a particular destination. For instance, sailors and pilots often utilise these functions to calculate the angle of elevation to a star or planet, and subsequently determine their path or direction. When the coordinates of the starting point and destination are known, arctan can be used to calculate the bearing or azimuthal angle, which is the angle between the north and the line from the starting point to the destination. This angle, crucial for navigation, helps in setting the course direction accurately, ensuring that the vehicle or individual reaches the desired location efficiently.

Inverse trigonometric functions have a profound relationship with complex numbers, particularly through Euler's formula, which states that e^(ix) = cos(x) + i sin(x), where i is the imaginary unit. The arctan function, for instance, can be used to find the argument of a complex number. If z is a complex number such that z = x + yi, where x and y are real numbers, then the argument of z, denoted arg(z), can be found using arctan as arg(z) = arctan(y/x). Furthermore, the arcsin and arccos functions can be used to express complex numbers in trigonometric form, which is particularly useful in operations like multiplication and division of complex numbers, as well as in finding their powers and roots.