Proving Euler's Formula: A Step-by-Step Guide

In summary, Euler's formula, also known as Euler's identity, is a mathematical equation connecting the exponential function and trigonometric functions. It has many applications in various fields of science, including physics, engineering, and computer graphics. Its proof involves using the power series expansion of the exponential function and trigonometric identities, and it can be understood by anyone with basic knowledge of algebra and trigonometry.
  • #1
mathhelp123
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Hi. I am trying to prove that (2sin^4x(2cos^2x+1))/3 = [sin^2(2x)/6-1/3][cos(2x)].

I tried fixing the right side, changing cos2x to 1 - 2sin^2x, and I went all the way to

(4sin^2x-4sin^4x-2-8sin^4x+8sin^6x+4sin^2x)/6.

I am clueless on how to continue the proof. Please help. Thanks.
 
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  • #2
Your notation isn't very clear but I think this may help:

[tex]e^{4 i \theta} = \left( e^{i \theta} \right)^4[/tex]

and apply Euler's formula to the individual exponentials.
 

1. What is Euler's formula?

Euler's formula, also known as Euler's identity, is a mathematical equation that relates the exponential function with trigonometric functions. It is written as e^(ix) = cos(x) + i*sin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is an angle in radians.

2. Why is Euler's formula important?

Euler's formula is important because it connects two fundamental branches of mathematics - calculus and geometry. It also has many applications in physics, engineering, and other fields of science.

3. What is the proof of Euler's formula?

The proof of Euler's formula involves using the power series expansion of the exponential function and the trigonometric identities for sine and cosine. It can also be derived using complex analysis and the Taylor series expansion of the exponential function.

4. Can anyone understand Euler's formula?

Yes, anyone with a basic knowledge of algebra and trigonometry can understand the concept of Euler's formula. However, the proof and its applications may require a deeper understanding of calculus and complex analysis.

5. How is Euler's formula used in real life?

Euler's formula has various real-life applications, including in electrical engineering, signal processing, and quantum mechanics. It is also used in computer graphics and animation to create smooth and realistic motion.

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