Absolute value of trigonometric functions of a complex number

In summary, the conversation discusses finding the real and imaginary parts of a complex number given by ##z=\sin(x+iy)##. The real part is ##\sin x \cosh y## and the imaginary part is ##\cos x \sinh y##. The conversation then moves on to simplify the absolute value formula for this complex number, which results in ##|z|=\sqrt{\sin^2 {x}+\sinh^2 {y}}##. An identity is used to eliminate ##\cosh^2 y## and the work is shown to arrive at the simplified form of the absolute value.
  • #1
agnimusayoti
240
23
Homework Statement
Find absolute value of ##\sin (x-iy)##.
Relevant Equations
$$\sin z=\frac {e^{iz}-e^{-iz}}{2i}$$
If ##z=x+iy## then, absolute value of this complex number is ##|z|=\sqrt {x^2+y^2}##
So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then
1. Real part: ##\sin x \cosh y##
2. Imaginary part: ##\cos x \sinh y##
If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }##
How to simplify that answer to ##|z|=\sqrt{\sin^2 {x}+\sinh^2 {y}}##?

Thanks
 
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  • #2
##\sin^2 + \cos^2 = 1##
 
  • #3
I do not understand how could ##\cosh^2 \cos^2## eliminated?
 
  • #4
Please show your work. If not it is impossible to know where you have gotten ##\cosh^2 \cos^2## from.
 
  • #5
You want to eliminate cos and cosh at some point. Do it right away.
$$\cos^2(x)=1-\sin^2(x)$$
$$\cosh^2(y)=1+\sinh^2(y)$$
 
  • #6
Uh there it is. Thanks!
 
  • #7
Orodruin said:
Please show your work. If not it is impossible to know where you have gotten ##\cosh^2 \cos^2## from.
I mean ##cosh^2 y## at ##sin^x cosh^2y##
 
  • #8
Please show your work. All of it.
 
  • #9
1588683901844..jpg


1588683942675..jpg
 

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  • #10
That is some nice work and handwriting.
a nice identity in there
$$u^2(1+v^2)+(1-u^2)v^2=u^2+v^2$$
 
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Related to Absolute value of trigonometric functions of a complex number

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

The absolute value of a trigonometric function of a complex number is the distance between the origin and the point on the complex plane represented by the function. It is calculated by taking the square root of the sum of the squares of the real and imaginary parts of the complex number.

2. How is the absolute value of a trigonometric function of a complex number related to its modulus?

The absolute value of a trigonometric function of a complex number is equal to the modulus of the complex number, which represents its magnitude or length. This means that the absolute value of a trigonometric function of a complex number is always a positive real number.

3. Can the absolute value of a trigonometric function of a complex number be negative?

No, the absolute value of a trigonometric function of a complex number is always a positive real number. This is because it represents the distance between the origin and a point on the complex plane, which cannot be negative.

4. How is the absolute value of a trigonometric function of a complex number calculated?

The absolute value of a trigonometric function of a complex number is calculated by taking the square root of the sum of the squares of the real and imaginary parts of the complex number. This can also be expressed as the modulus of the complex number.

5. What is the significance of the absolute value of a trigonometric function of a complex number?

The absolute value of a trigonometric function of a complex number is important in understanding the properties and behavior of complex numbers. It is also used in various mathematical applications, such as in solving equations involving complex numbers and in visualizing complex functions on the complex plane.

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