How to calculate the trigonometric functions of a complex number?

To calculate the trigonometric functions of a complex number, use the formula z = x + iy.

To find the trigonometric functions of a complex number, we first need to convert it into polar form. This involves finding the modulus (r) and argument (θ) of the complex number. The modulus is the distance from the origin to the point representing the complex number in the complex plane, and the argument is the angle between the positive real axis and the line joining the origin to the point representing the complex number.

r = |z| = √(x^2 + y^2)

θ = arg(z) = tan^-1(y/x) (taking into account the quadrant in which the point lies)

Once we have found r and θ, we can use the following formulas to find the trigonometric functions:

sin(θ) = y/r

cos(θ) = x/r

tan(θ) = y/x

cosec(θ) = r/y

sec(θ) = r/x

cot(θ) = x/y

For example, let's find the trigonometric functions of the complex number z = 3 + 4i.

r = |z| = √(3^2 + 4^2) = 5

θ = arg(z) = tan^-1(4/3) = 53.13° (in the second quadrant)

sin(θ) = y/r = 4/5

cos(θ) = x/r = 3/5

tan(θ) = y/x = 4/3

cosec(θ) = r/y = 5/4

sec(θ) = r/x = 5/3

cot(θ) = x/y = 3/4

Therefore, the trigonometric functions of z = 3 + 4i are sin(θ) = 4/5, cos(θ) = 3/5, tan(θ) = 4/3, cosec(θ) = 5/4, sec(θ) = 5/3, and cot(θ) = 3/4.