How to calculate the tangent of a complex number?

To calculate the tangent of a complex number, use the formula tan(z) = (sin(2x) + i sinh(2y)) / (cos(2x) + cosh(2y)), where z = x + iy.

The tangent of a complex number can be found using the formula tan(z) = (sin(2x) + i sinh(2y)) / (cos(2x) + cosh(2y)), where z = x + iy. To use this formula, first find the values of x and y for the given complex number.

For example, let z = 2 + 3i. Then x = 2 and y = 3. Next, use the formula to find the tangent:

tan(z) = (sin(2x) + i sinh(2y)) / (cos(2x) + cosh(2y))
tan(2 + 3i) = (sin(4) + i sinh(6)) / (cos(4) + cosh(6))

Using the definitions of hyperbolic sine and cosine, this can be simplified to:

tan(2 + 3i) = (sin(4) + i (e^6 - e^-6) / 2) / (cos(4) + (e^6 + e^-6) / 2)

Evaluating the trigonometric functions and simplifying, the final answer is:

tan(2 + 3i) = -0.0037 + 1.0039i

For a deeper understanding of how trigonometric functions relate to complex numbers, explore our notes on the trigonometric form of complex numbers. Additionally, to effectively utilise trigonometric identities in calculations involving complex numbers, consider reviewing our trigonometric identities page. For further clarification of basic trigonometric ratios which are fundamental in understanding these concepts, refer to our trigonometric ratios guide.