Let's break down the problem step by step to find the magnitude of [tex]\( e^{ix} \)[/tex].
Given that [tex]\( z = x + iy \)[/tex], where [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are real numbers:
1. Euler's Formula:
Euler's formula states that [tex]\( e^{ix} = \cos(x) + i\sin(x) \)[/tex]. This is a fundamental result from complex analysis which allows us to express the exponential function of an imaginary argument in terms of trigonometric functions.
2. Magnitude of a Complex Number:
The magnitude (or absolute value) of a complex number [tex]\( a + ib \)[/tex] is given by [tex]\( \sqrt{a^2 + b^2} \)[/tex]. In our case, the complex number in question is [tex]\( \cos(x) + i\sin(x) \)[/tex].
3. Applying Euler's Formula:
So, using Euler's formula, the complex number [tex]\( e^{ix} \)[/tex] can be represented as [tex]\( \cos(x) + i\sin(x) \)[/tex].
4. Finding the Magnitude:
The magnitude of [tex]\( e^{ix} = \cos(x) + i\sin(x) \)[/tex] is:
[tex]\[
\left| e^{ix} \right| = \sqrt{(\cos(x))^2 + (\sin(x))^2}
\][/tex]
5. Using Pythagorean Identity:
From trigonometry, we know the Pythagorean identity which states that [tex]\( (\cos(x))^2 + (\sin(x))^2 = 1 \)[/tex].
6. Final Result:
Therefore, substituting this identity into our magnitude formula, we get:
[tex]\[
\left| e^{ix} \right| = \sqrt{1} = 1
\][/tex]
So, the magnitude of [tex]\( e^{ix} \)[/tex] is always [tex]\( 1 \)[/tex] for any real value of [tex]\( x \)[/tex].
In conclusion, the magnitude [tex]\(\left|e^{ix}\right|\)[/tex] is:
[tex]\[
1.0
\][/tex]
