Solving for z in the Equation tan z = 1 + 2i

In summary, to find the values of tan-1(1+2i), we can use the fact that tan-1z = (i/2)log((i+z)/(i-z)). By substituting the given values, we get (i/2)log((1+3i)/(-i-1)). Simplifying the argument to the log function, we get (i/2)(log(1+3i)-log(-1-i)). We are not "solving" these expressions since they are not equations, but rather simplifying them. Finally, to find the value of z, we can write the equation \tan z =1+2i=-i\frac{e^{iz}-e^{-
  • #1
ver_mathstats
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Homework Statement
Find values of tan^-1(1+2i)
Relevant Equations
tan^-1(1+2i)
Find the values of tan-1(1+2i).

We can use the fact: tan-1z = (i/2)log((i+z)/(i-z)).

Then with substitutions we have (i/2)log((1+3i)/(-i-1)).

Then I think the next step would be (i/2)(log(1+3i)-log(-1-i)).

Do we then just proceed to solve log(1+3i) and log(-1-i)? I'm just a little confused what the next step should be, any help is appreciated, thank you.
 
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  • #2
ver_mathstats said:
Then with substitutions we have (i/2)log((1+3i)/(-i-1)).

Then I think the next step would be (i/2)(log(1+3i)-log(-1-i)).
Instead, simplify the argument to the log function. That fraction simplifies to -2 - i .
ver_mathstats said:
Do we then just proceed to solve log(1+3i) and log(-1-i)?
You're not "solving" these expressions since they're not equations.
 
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  • #3
Mark44 said:
Instead, simplify the argument to the log function. That fraction simplifies to -2 - i .

You're not "solving" these expressions since they're not equations.
Sorry, I miswrote that.
 
  • #4
Let z be the value we want, I would write the equation
[tex] \tan z =1+2i=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}[/tex]
to get z.
 
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FAQ: Solving for z in the Equation tan z = 1 + 2i

1. What is the definition of a complex trigonometric function?

A complex trigonometric function is a mathematical function that involves complex numbers and uses trigonometric functions such as sine, cosine, and tangent.

2. How do you find the value of a complex trigonometric function?

To find the value of a complex trigonometric function, you can use the Euler's formula, which expresses a complex number in terms of sine and cosine. You can also use a calculator or computer program to evaluate the function.

3. What is the difference between a real and complex trigonometric function?

A real trigonometric function only involves real numbers, whereas a complex trigonometric function involves complex numbers. Real trigonometric functions can be graphed on a two-dimensional plane, while complex trigonometric functions require a three-dimensional graph.

4. How can complex trigonometric functions be applied in real life?

Complex trigonometric functions have many applications in fields such as engineering, physics, and mathematics. They are used to model and solve problems involving alternating currents, electromagnetic waves, and oscillations.

5. Are there any special properties of complex trigonometric functions?

Yes, complex trigonometric functions have several special properties, such as periodicity, symmetry, and analyticity. They also follow similar rules and identities as real trigonometric functions, but with some modifications due to the involvement of complex numbers.

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