Understanding the Formulas for the Inverse Tangent Function

The inverse tangent, also known as the arctangent, is a fundamental trigonometric function denoted as tan^{-1}(y/x). It is used to determine the angle θ given the ratio of the opposite and adjacent sides of a right triangle, i.e., tan(θ) y/x.

The Basic Formula for Inverse Tangent

The basic formula for the inverse tangent function is:

tan^{-1}(y/x) θ, where θ is the angle such that tan(θ) y/x.

In programming languages and calculators, the function is often implemented as atan(y/x) or atan2(y, x). The latter takes into account the signs of both y and x to determine the correct quadrant for the angle θ.

The Taylor Series Expansion for θ

The inverse tangent function can also be expressed using the Taylor series expansion. When x 1 and y 1, the formula is:

tan^{-1}(1/x) 1 - x2 x4 - x6 ... $$sum_{i0}^{infty} (-1)^i frac{x^{2i 1}}{2i 1}$$

This series converges for all x such that |x|

Special Cases of Inverse Tangent

There are several special cases to consider for the inverse tangent function:

When x 0, the angle θ is 90 degrees or 2n(π/2), depending on the value of y.

When y 0, the angle θ is 0 degrees or 2nπ, where n is an integer.

If x 0 and y 0, the angle θ is in the first quadrant.

If x 0 and y 0, the angle θ is in the second quadrant.

If x 0 and y 0, the angle θ is in the third quadrant.

If x 0 and y 0, the angle θ is in the fourth quadrant.

For complex numbers, the inverse tangent function can be expressed using the inverse hyperbolic tangent:

Let z y/x, then θ arctanh(z) (e^z - e^{-z}) / (e^z e^{-z})

The ln Formulation for Inverse Tangent

The inverse tangent function can also be expressed using a logarithmic function:

arctan(z) 0.5 ln((1 - iz) / (1 iz)), where i is the imaginary unit and ln is the natural logarithm.

This formulation is particularly useful in complex analysis and related fields.

Conclusion

The inverse tangent function, or arctangent, is a versatile and important mathematical tool, with applications ranging from geometry to complex analysis. Understanding its various forms and properties can greatly enhance problem-solving capabilities in a wide range of fields.