Find Roots of sin z: Solutions & Explanations

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In summary, the roots of the equation $\sin z = 0$ are $n\pi$, where $n$ is an integer. The roots of the equation $\sin(x+iy)=0$ are $(n\pi, 0)$, where $n$ is an integer.
  • #1
ognik
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Looking for someone to check my working & answers please. Problem is 'find all the zeros of sin z'

A) sin z = sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y)
Roots are when sin(x)cosh(y) = 0 = cos(x)sinh(y)

$If \: sinh(y)=0, then \: cosh(y)=1 \: (cosh^2 - sinh^2=1) $
$ \therefore sin(x) = 0, \therefore x = n\pi, n=0,\pm1,\pm2 ... $
$ y = arcsinh(0) = 0 $
$ \therefore $ 1st root is $ (0,n\pi) $

B) I also would like to do it using exponents:
$ sin z = \frac{1}{2i}\left(e^{z}-e^{-z}\right) $
$ \therefore \left(e^{z}-e^{-z}\right) = 0 $
$ \therefore e^{z} = e^{-z}, \: \therefore x+iy = -(x+iy)$
This just gives x = 0 = y which seems trivial?
Thanks in advance.
 
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  • #2
ognik said:
Looking for someone to check my working & answers please. Problem is 'find all the zeros of sin z'

A) sin z = sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y)
Roots are when sin(x)cosh(y) = 0 = cos(x)sinh(y)

$If \: sinh(y)=0, then \: cosh(y)=1 \: (cosh^2 - sinh^2=1) $
$ \therefore sin(x) = 0, \therefore x = n\pi, n=0,\pm1,\pm2 ... $
$ y = arcsinh(0) = 0 $
$ \therefore $ 1st root is $ (0,n\pi) $

B) I also would like to do it using exponents:
$ sin z = \frac{1}{2i}\left(e^{z}-e^{-z}\right) $
$ \therefore \left(e^{z}-e^{-z}\right) = 0 $
$ \therefore e^{z} = e^{-z}, \: \therefore x+iy = -(x+iy)$
This just gives x = 0 = y which seems trivial?
Thanks in advance.

Hi ognik,

(A) is fine, although saying the 1st root is $ (0,n\pi) $ isn't proper.
It should say something like "the roots are $n\pi$ where $n$ is an integer".
That's because it's not only the 1st root, but all roots.
And $ (0,n\pi) $ either denotes the open interval from $0$ to $n\pi$, or the point with x-coordinate $0$ and y-coordinate $n\pi$.
Neither is a root.I'm afraid that in (B) the formula for $\sin z$ is wrong.
It should be $ \sin z = \frac{1}{2i}\left(e^{iz}-e^{-iz}\right) $

Continuing, we get:
$$ e^{i(x+iy)} = e^{-i(x+iy)} $$
Now, when dealing with complex numbers, we can't just take the natural logarithm, because it is a multivalued function.
Properly, we need to account for the fact that the function $e^{ix}$ has a period of $2\pi$, which results in:
$$ i(x+iy) = -i(x+iy + 2\pi n)
\Rightarrow x+iy = -x -iy - 2\pi n
\Rightarrow 2(x + \pi n) + 2iy = 0
\Rightarrow x = -\pi n \wedge y = 0
$$
 
  • #3
Thanks ILS, again a careless mistake :-(

I am still coming to terms with some of the variable meanings in math, I had intended $ (n \pi, 0) $ to represent the values (x, y), not an interval or coordinate; so should I rather have said the roots are $ n \pi + i0, n=0, \pm1,... $?
Then - if we were talking about the roots of a multi-varied (say x,y,z), real equation - how would I properly write the roots?
 
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  • #4
ognik said:
Thanks ILS, again a careless mistake :-(

I am still coming to terms with some of the variable meanings in math, I had intended $ (n \pi, 0) $ to represent the values (x, y), not an interval or coordinate; so should I rather have said the roots are $ n \pi + 0i, n=0, \pm1,... $?
Then - if we were talking about the roots of a multi-varied (say x,y,z), real equation - how would I properly write the roots?

In an equation with real $x,y,z$, we have roots that are elements of $\mathbb R^3$. Those elements are represented with $(x,y,z)$.
In an equation with complex numbers $z \in \mathbb C$ the roots are indeed like $ n \pi + i0$ or just $n\pi$.
In an equation with real $x,y$, we can have roots like $(n\pi, 0)$, which are elements of $\mathbb R^2$.

So your equation $\sin z = 0$ has roots $n\pi$, while the equation $\sin(x+iy)=0$ has roots $(n\pi, 0)$.
 

FAQ: Find Roots of sin z: Solutions & Explanations

What does it mean to find the roots of sin z?

Finding the roots of sin z means determining the values of z that satisfy the equation sin z = 0. These values are known as the roots or solutions of the equation.

Why is it important to find the roots of sin z?

Knowing the roots of sin z allows us to solve various mathematical problems involving trigonometric functions, such as finding the period or amplitude of a sine wave. It also has applications in physics, engineering, and other fields.

How do you find the roots of sin z?

To find the roots of sin z, we can use the fact that sin z = 0 when z is a multiple of π. Therefore, the roots of sin z are z = nπ, where n is any integer.

Are there any other methods for finding the roots of sin z?

Yes, there are other methods for finding the roots of sin z, such as using the graph of the sine function or using trigonometric identities. These methods may be more efficient for certain equations or situations.

Can there be more than one root of sin z?

Yes, there can be infinitely many roots of sin z since the sine function is periodic with a period of 2π. This means that for any integer n, z = nπ is a root of sin z. However, we typically only consider the principal or smallest positive root in certain contexts.

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