Euler's Forumla, Trig Addition, and Equating Coefficients

In summary, the conversation discusses the use of Euler's formula to quickly obtain the addition formulas for sine and cosine in precalculus. The concept of equating coefficients with respect to the imaginary part i is also mentioned, with a question about how this can be proven to be true. The conversation ends with a clarification of equating coefficients and how it applies to complex numbers.
  • #1
elarson89
20
0
One can look into any precalculus book and find a proof of the addition formulas of sine and cosine. Though as most are aware there is a quick way to get the formulas by using Euler's Formula. But to get the formulas by eulers formula, you must equate coefficients with respect to the imaginary part i.

My question is this, equating coefficients was taught to be used for polynomials, because a set of coefficients uniquely determines a polynomial. How can you show the same is true with respect to i? Yes it looks very intuitive, but I'm wondering if there's something a little more powerful than that.
 
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  • #2
What do you mean by "equate coefficients"? Maybe you could show us what theorem you're trying to prove to make it clearer.
 
  • #3
Theres, only one addition formula for sine... and then there is the euler's formula e^ix=... and only one way to equate coefficients... I would write them out but I don't know how to embed tex.
 
  • #4
if you have a + bi = c + di, then by the definition of equality of complex numbers you must have a = c and b = d. so you can equate the real and imaginary parts, if that's what you're asking.
 

What is Euler's Formula?

Euler's Formula is a mathematical equation that relates the trigonometric functions sine and cosine to the complex exponential function.

How is Euler's Formula used?

Euler's Formula is used in many areas of mathematics, physics, and engineering, including signal processing, differential equations, and Fourier analysis.

What is the Trig Addition Formula?

The Trig Addition Formula is a set of equations that allow you to find the sine, cosine, tangent, and other trigonometric functions of the sum or difference of two angles.

Why is the Trig Addition Formula useful?

The Trig Addition Formula is useful for simplifying complicated trigonometric expressions, solving trigonometric equations, and finding relationships between angles and trigonometric functions.

What does it mean to equate coefficients?

Equating coefficients means setting the coefficients, or numbers in front of variables, in two equations equal to each other. This is often used to solve systems of equations or to find unknown variables.

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