Finding Solutions to the Sinh Equation in Complex Numbers

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In summary: Can you please help me out with finding the solution for k other than 0?In summary, in equation 1, cos(y) = 0 or \sinh (x) = 0, and x = 0 and y = \frac{\pi }{2} + 2k\pi. However, for k other than 0, the solution is not given.
  • #1
Benny
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Hello, can someone please help me out with the following question?

Express [tex]\sinh \left( {x + yi} \right)[/tex] in terms of [tex]\sinh (x)[/tex], [tex]\cosh(x)[/tex], [tex]\cos(y)[/tex] and [tex]\sin(y)[/tex]. Hence find all solutions [tex]z \in C[/tex] of the equation [tex]\sinh(z) = i[/tex].

I have little idea as to how to do this question. Here is my working.

[tex]\sinh \left( {x + yi} \right) = \sinh \left( x \right)\cosh \left( {iy} \right) + \cosh \left( x \right)\sinh \left( {iy} \right)[/tex]

[tex] = \sinh \left( x \right)\left( {\frac{{e^{iy} + e^{ - iy} }}{2}} \right) + \cosh \left( x \right)\left( {\frac{{e^{iy} - e^{- iy} }}{2}} \right)[/tex]

[tex]{\rm = sinh}\left( {\rm x} \right)\left( {\frac{{e^{iy} + e^{iy} }}{2}} \right) + i\cosh \left( x \right)\left( {\frac{{e^{iy} - e^{ - iy} }}{{2i}}} \right)[/tex]

[tex] = \sinh \left( x \right)\cos \left( y \right) + i\cosh \left( x \right)\sin \left( y \right)[/tex]

So [tex]\sinh \left( z \right) = i \Rightarrow \sinh \left( x \right)\cos \left( y \right) + i\cosh \left( x \right)\sin \left( y \right) = i[/tex]

Equating real and imaginary parts:

[tex]\sinh \left( x \right)\cos \left( y \right) = 0...(1)[/tex]

[tex]\cosh \left( x \right)\sin \left( y \right) = 1...(2)[/tex]

At this point I am unsure of how to proceed. The equations do not look solvable, directly anyway. So I decided to start off with equation 1 and see where that would lead. Here is what I have done.

[tex]\sinh \left( x \right)\cos \left( y \right) = 0[/tex]

[tex] \Rightarrow \cos (y) = 0[/tex] or [tex]\sinh (x) = 0[/tex]

[tex]\frac{{e^x - e^{ - x} }}{2} = 0 \Rightarrow x = 0[/tex] and [tex]y = \frac{{n\pi }}{2}[/tex] where n is an odd ineger.

Now since equations (1) and (2) need to be satisfied simultaneously then x is necessarily equal to zero and equation (2) reduces to [tex]\sin \left( y \right) = 1[/tex] which has solutions: [tex]y = \frac{\pi }{2} + 2k\pi[/tex] where k is an integer.

I need y to satisfy both equations (1) and (2) so I take y to be the 'least general' answer so that [tex]y = \frac{\pi }{2} + 2k\pi[/tex].

So I end up with [tex]\sinh \left( z \right) = i \Leftrightarrow z = \left( {\frac{\pi }{2} + 2k\pi } \right)i[/tex]. I'm not sure if my method is right. I pretty much got stuck at the simultaneous equations part. Can someone please hep me out with this question?
 
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  • #2
It looks okay to me...

You may have tried to check whether

[tex] \sinh\left(\frac{i\pi}{2}+2ki\pi\right)=i [/tex]


Daniel.
 
  • #3
Thanks for your response dextercioby. I have checked that my solution works for k = 0 but not for any other values.
 

Related to Finding Solutions to the Sinh Equation in Complex Numbers

1. What is the cos and cosign equation?

The cos and cosign equations are mathematical formulas that describe the relationship between the sides and angles of a right triangle. The cosine (cos) function is used to determine the ratio of the adjacent side to the hypotenuse, while the cosine inverse (cos⁻¹) function is used to find the measure of an angle given the ratio of its sides.

2. How do I solve a cos and cosign equation?

To solve a cos or cosign equation, you will need to know the values of at least two sides or angles of a right triangle. You can then use trigonometric identities and algebraic manipulation to find the missing values. It is also helpful to have a calculator with trigonometric functions to assist in calculations.

3. What are some common applications of cos and cosign equations?

Cos and cosign equations have many practical applications in fields such as engineering, physics, and astronomy. They are used to calculate forces, distances, and angles in various real-world scenarios involving right triangles, such as determining the height of a building or the distance between two objects.

4. Can cos and cosign equations be used for non-right triangles?

No, cos and cosign equations are specific to right triangles and cannot be used for non-right triangles. However, there are other trigonometric functions such as sine and tangent that can be used for non-right triangles.

5. How can I improve my understanding of cos and cosign equations?

To improve your understanding of cos and cosign equations, it is important to practice solving various types of problems and familiarize yourself with the different identities and formulas. You can also seek help from a tutor or online resources, and use visual aids such as diagrams to better understand the concepts.

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