# Expressing sum of sines and cosines as a complex exponential

• MuIotaTau
In summary, there is a way to turn the function f(x) = A cos (x) + B sin (x) into an expression of the form F(x) = C e^{i(x + \phi)}, by using Euler's formula to express F(x) as the sum of cos and sin, and then finding relations between the constants C and ϕ and the original constants A and B.
MuIotaTau
If I'm given a function ##f(x) = A cos (x) + B sin (x)##, is there any way to turn this into an expression of the form ##F(x) = C e^{i(x + \phi)}##? I know how to use Euler's formula to turn this into ## \alpha e^{i(x + \phi)} + \beta e^{-i(x + \phi)}##, but is there a way to incorporate the second term into the first somehow, maybe with a change in the constants?

You can use Euler's formula to express F(x) as sum of cos and sin and then find relations for the constants C and ϕ as function of A and B.

## What is the formula for expressing the sum of sines and cosines as a complex exponential?

The formula is e^(iθ)=cosθ+isinθ. This is known as Euler's formula.

## Why is it useful to express the sum of sines and cosines as a complex exponential?

Expressing complex trigonometric functions as a single exponential function can make calculations and proofs in mathematics and physics much simpler and more efficient.

## How does expressing the sum of sines and cosines as a complex exponential relate to the concept of a unit circle?

The complex exponential form can be thought of as a way to represent points on the unit circle in the complex plane. The angle θ in the formula corresponds to the angle between the positive real axis and the point on the unit circle.

## Can the formula for expressing the sum of sines and cosines as a complex exponential be extended to include more terms?

Yes, the formula can be extended to include any number of terms. For example, e^(3iθ)=cos3θ+isin3θ. This is known as De Moivre's formula.

## Are there any practical applications of expressing the sum of sines and cosines as a complex exponential?

Yes, this formula is commonly used in electrical engineering and signal processing to analyze and manipulate signals that are represented in the frequency domain.

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