Derive Sin(x+y) & Cos(x+y) Using Euler's Formula

In summary, there are two ways to derive the sin(x+y) and cos(x+y) expansions. One way is by using Euler's formula, which can be tedious but straightforward. Another way is by using 2x2 rotation matrices, which is equivalent to using Euler's formula but in R^2 instead of complex numbers. There is a short proof available on Wikipedia, although it is not a completed proof.
  • #1
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Can anyone tell me how to derive the sin(x+y) and cos(x+y) expansions? The ones that are like cos x sin y or sin y cos x + other stuff?
Preferrably, could this be derived with Euler's formula alone? Or something not too geometric? (All those OAs and OBs and XBs and XYs on geometric diagrams confuse me too much to follow)
Thank you.
 
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  • #2
You could use Euler's formula. It is tedious, but straightforward.
 
  • #3
Another way is to use the 2X2 rotation matrices R([itex]\theta[/itex]), which have R(x)R(y)=R(x+y). This is equivalent to using Euler's formula, only you're working in R^2 instead of the complex numbers.
 
  • #4
There's a short proof at wikipedia, you can view it at the end of this page.
It is, however, not a completed proof, but you can get some ideas about proving it. :)
 

1. What is Euler's Formula?

Euler's Formula is a mathematical equation that relates the complex exponential function to the 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. How do you use Euler's Formula to derive Sin(x+y)?

To derive Sin(x+y) using Euler's Formula, we first write it as e^(i(x+y)). Then, using the properties of exponents, we can expand it as e^(ix)*e^(iy). This can be rewritten as (cos(x)+i*sin(x))*(cos(y)+i*sin(y)). Expanding this further, we get cos(x)cos(y) - sin(x)sin(y) + i(cos(x)sin(y) + sin(x)cos(y)). Finally, equating the imaginary parts, we get Sin(x+y) = sin(x)cos(y) + cos(x)sin(y).

3. How do you use Euler's Formula to derive Cos(x+y)?

Similar to deriving Sin(x+y), we can use Euler's Formula to derive Cos(x+y) by writing it as e^(i(x+y)). Then, using the same steps as above, we get Cos(x+y) = cos(x)cos(y) - sin(x)sin(y).

4. Can Euler's Formula be used to derive other trigonometric identities?

Yes, Euler's Formula can be used to derive other trigonometric identities such as Tan(x+y), Cot(x+y), Sec(x+y), Csc(x+y), and their inverse functions. It can also be used to prove De Moivre's Theorem for complex numbers.

5. What are some practical applications of Euler's Formula?

Euler's Formula has many practical applications in fields such as physics, engineering, and signal processing. It is used to simplify complex calculations involving trigonometric functions, as well as in the analysis of alternating current circuits, vibrations, and waves. It is also used in Fourier analysis, which is essential in digital signal processing and data compression.

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