Sinhz = 0 iff z=n(pi)i (n=0, +/- 1, +/- 2 ) ~ a question about this?

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In summary, the question is asking for details to show that the zeros of sinhz and coshz, as mentioned in statements (14) and (15), can be found by substituting the given values into the formulas for sinh and cosh. The values given are z=n(pi)i and z=(pi/2 + n(pi)), respectively, where n is any integer. The formulas for sinh and cosh are sinh(z) = [e^z - e^(-z) ] /2 and cosh(z) = [e^z + e^(-z)]/2, and the identities -sin(iz) = sinh(z) and cos(iz) = cosh(z) are also provided.
  • #1
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Homework Statement


I have this weird question in my textbook I'm not even sure which part is the question and what i need to do...

It says

"Give details showing that the zeros of sinhz and coshz are as in statements (14) and (15) "


Homework Equations



(14) sinhz = 0 iff z=n(pi)i (n=0, +/- 1, +/- 2...)

(15) coshz = 0 iff z=(pi/2 + n(pi)) (n=0, +/- 1, +/- 2...)

-sin(iz) = sinh(z)
cos(iz) = cosh(z)
sinh(z) = [e^z - e^(-z) ] /2
cosh(z) = [e^z + e^(-z)]/2

The Attempt at a Solution



I haven't made a start on the question because I don't understand the words and what it wants me to calc? :grumpy:
 
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  • #2
Substitute the given values into your formulas for sinh and cosh and show they are zero. The words look pretty clear to me. I think you've forgotten an i in your cosh zeros.
 
  • #3


Hello,

I can understand your confusion with this question. Let me try to break it down for you.

First, let's define the terms in the equations (14) and (15):

- Sinhz: This is the hyperbolic sine function, also written as sinh(z). It is defined as sinh(z) = (e^z - e^(-z))/2.

- Coszh: This is the hyperbolic cosine function, also written as cosh(z). It is defined as cosh(z) = (e^z + e^(-z))/2.

- Zeros: These are the values of z for which the function is equal to 0.

Now, let's look at the statements (14) and (15):

- Statement (14) says that the hyperbolic sine function, sinh(z), is equal to 0 if and only if z is equal to n(pi)i for n = 0, +/-1, +/-2, etc. This means that if z is a multiple of pi times i (imaginary number), then sinh(z) will be equal to 0.

- Statement (15) says that the hyperbolic cosine function, cosh(z), is equal to 0 if and only if z is equal to (pi/2 + n(pi)) for n = 0, +/-1, +/-2, etc. This means that if z is equal to (pi/2 + n(pi)), then cosh(z) will be equal to 0.

Now, for the question, it is asking you to give some details or explanations to show that these statements are true. In other words, you need to prove that these statements are correct. This can be done by using the properties of the hyperbolic functions and the definitions of sinh(z) and cosh(z) given above.

For example, to prove statement (14), you can start by substituting z = n(pi)i into the definition of sinh(z). This will give you sinh(n(pi)i) = (e^(n(pi)i) - e^(-n(pi)i))/2. Using the properties of exponential functions, you can simplify this to sinh(n(pi)i) = (e^(n(pi)i) - 1)/2. Since e^(n(pi)i) = cos(n(pi)) + i*sin(n(pi)), we can see that if n is an integer, then cos(n(pi)) = 1 and sin
 

1. What does "Sinhz = 0 iff z=n(pi)i (n=0, +/- 1, +/- 2 )" mean?

The equation "Sinhz = 0 iff z=n(pi)i (n=0, +/- 1, +/- 2 )" means that the hyperbolic sine function will equal 0 when the complex number z is equal to a multiple of pi multiplied by the imaginary unit i, where n is any integer including 0, +/- 1, and +/- 2. In other words, the solution to this equation will result in a set of complex numbers that satisfy the equation.

2. What is the significance of the complex number z=n(pi)i (n=0, +/- 1, +/- 2 ) in this equation?

The complex number z=n(pi)i (n=0, +/- 1, +/- 2 ) is significant because it represents the values for which the hyperbolic sine function is equal to 0. This equation helps to determine the roots of the hyperbolic sine function, which can have important applications in mathematics and physics.

3. How does this equation relate to trigonometry?

This equation relates to trigonometry because it involves the use of the imaginary unit i, which is commonly used in trigonometric functions. Additionally, the values of n(pi) in the equation can be thought of as angles in radians, similar to how angles are measured in trigonometry.

4. Can this equation be used to solve for other trigonometric functions?

Yes, this equation can be used to solve for other trigonometric functions such as the hyperbolic cosine and tangent functions. By substituting the values for z in the original equation, we can find the roots of these functions as well.

5. What are some practical applications of this equation?

This equation has various practical applications in fields such as physics, engineering, and mathematics. It can be used to solve for the roots of the hyperbolic sine function, which can then be used to model various real-world phenomena such as electromagnetic waves, oscillations, and electrical circuits.

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