How to calculate the cosine of a complex number?

The cosine of a complex number can be calculated using the exponential form of the complex number.

To calculate the cosine of a complex number, we first need to express the complex number in exponential form. Let z = x + iy be a complex number, where x and y are real numbers. Then, we can express z in exponential form as z = re^(iθ), where r = |z| is the modulus of z and θ is the argument of z.

The cosine of z can then be calculated using the formula cos(z) = (e^(iz) + e^(-iz))/2. To see why this formula works, we can use Euler's formula, which states that e^(ix) = cos(x) + i sin(x) for any real number x.

Substituting z = x + iy into the formula for cos(z), we get:

cos(z) = (e^(i(x+iy)) + e^(-i(x+iy)))/2
= (e^(ix)e^(-y) + e^(-ix)e^(y))/2
= (cos(x)e^(-y) + cos(x)e^(y))/2 + i(sin(x)e^(-y) - sin(x)e^(y))/2
= cos(x)cosh(y) + i sin(x)sinh(y)

Therefore, the cosine of a complex number z = x + iy is given by cos(z) = cos(x)cosh(y) + i sin(x)sinh(y).