Inverse tangent of a complex number

• bonfire09
In summary, to find ##\tan^{-1}(2i)##, one can use the formula ##\tan{ix} = itanh{x}## or solve the equation ##-3=e^{-2zi}## by using the fact that ##\tan{z}=2i##.
bonfire09

Homework Statement

I have to find ##\tan^{-1}(2i)##.

The Attempt at a Solution

So far I have ##\tan^{-1}(2i)=z\iff tan z= 2i\iff \dfrac{sin z}{cos z}=2i ##. From here I get that
##-3=e^{-2zi}##. I do no know how to take it further to get ##z=i\dfrac{\ln 3}{2}+(\dfrac{\pi}{2}+\pi n)##. Should I use natural logarithms but the problem is that I have a ##-3## which won't allow me to take the natural log of both sides. Any help would be great thanks.

bonfire09 said:
I get that
##-3=e^{-2zi}##.
I get -(3+4i)/5 for that.
I have a ##-3## which won't allow me to take the natural log of both sides.
The log of a negative number is fine when complex answers are allowed. It's only 0 that has no log in the complex plane.

bonfire09 said:

Homework Statement

I have to find ##\tan^{-1}(2i)##.

The Attempt at a Solution

So far I have ##\tan^{-1}(2i)=z\iff tan z= 2i\iff \dfrac{sin z}{cos z}=2i ##. From here I get that
##-3=e^{-2zi}##.

The solution is as simple as using ##\tan{ix} = itanh{x}##.

But you can use your method and do the algebra, it's just a little more work. Comes to the same answer.

Last edited:

What is the inverse tangent of a complex number?

The inverse tangent of a complex number is a mathematical operation that gives the angle in radians whose tangent is equal to the given complex number. It is denoted as tan-1(z), where z is the complex number.

How do you calculate the inverse tangent of a complex number?

To calculate the inverse tangent of a complex number, you can use the formula tan-1(z) = (1/2i) * [ln(1-iz) - ln(1+iz)], where i is the imaginary unit √(-1). Alternatively, you can use a scientific calculator or computer software to find the inverse tangent of a complex number.

What is the range of inverse tangent of a complex number?

The range of inverse tangent of a complex number is from -π/2 to π/2, or from -90° to 90° in degrees. This means that the output of the inverse tangent function will always be within this range, regardless of the input complex number.

What is the principal value of inverse tangent of a complex number?

The principal value of inverse tangent of a complex number is the value within the range of -π/2 to π/2. This is considered as the main or most important value of the inverse tangent function, and it is often denoted as tan-1 or arctan.

What are the properties of the inverse tangent of a complex number?

The inverse tangent of a complex number has the following properties:
1. tan-1(0) = 0
2. tan-1(∞) = π/2
3. tan-1(-∞) = -π/2
4. tan-1(z) = π/2 - tan-1(1/z)
5. tan-1(-z) = -tan-1(z)
6. tan-1(z*) = tan-1(z), where z* is the complex conjugate of z.

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