show using euler's formula  A cos(kx − ωt) + B sin(kx − ωt) (5) can be equivalently written as C cos(kx − ωt + φ). Find C and φ in terms of A and B.

We want to express the trigonometric sum \( A \cos(kx - \omega t) + B \sin(kx - \omega t) \) in the form \( C \cos(kx - \omega t + \varphi) \), where \( C \) and \( \varphi \) are in terms of \( A \) and \( B \).

We can use Euler's formula to achieve this. Euler's formula states that:

\[
e^{i\theta} = \cos(\theta) + i \sin(\theta)
\]

### Step 1: Express the left-hand side using complex exponentials

We rewrite \( A \cos(kx - \omega t) + B \sin(kx - \omega t) \) using Euler's formula:

\[
\cos(kx - \omega t) = \frac{e^{i(kx - \omega t)} + e^{-i(kx - \omega t)}}{2}
\]
\[
\sin(kx - \omega t) = \frac{e^{i(kx - \omega t)} - e^{-i(kx - \omega t)}}{2i}
\]

Thus,

\[
A \cos(kx - \omega t) + B \sin(kx - \omega t) = A \left( \frac{e^{i(kx - \omega t)} + e^{-i(kx - \omega t)}}{2} \right) + B \left( \frac{e^{i(kx - \omega t)} - e^{-i(kx - \omega t)}}{2i} \right)
\]

After combining terms, this becomes:

\[
\frac{A - iB}{2} e^{i(kx - \omega t)} + \frac{A + iB}{2} e^{-i(kx - \omega t)}
\]

### Step 2: Use polar form to simplify

Now, express \( A - iB \) as a complex number in polar form. This gives us the magnitude \( C \) and phase \( \varphi \) of the complex number \( A - iB \):

\[
A - iB = C e^{-i\varphi}
\]
where \( C = \sqrt{A^2 + B^2} \) is the magnitude, and \( \varphi \) is the phase, given by:

\[
\varphi = \tan^{-1}\left(\frac{B}{A}\right)
\]

Thus, we can rewrite the expression as:

\[
C \cos(kx - \omega t + \varphi)
\]

### Step 3: Conclusion

The expression \( A \cos(kx - \omega t) + B \sin(kx - \omega t) \) can indeed be written as \( C \cos(kx - \omega t + \varphi) \), where:

\[
C = \sqrt{A^2 + B^2}
\]
\[
\varphi = \tan^{-1}\left(\frac{B}{A}\right)
\]

Let me know if you'd like more details or examples! 

Here are 5 related questions:
1. How does Euler's formula simplify trigonometric expressions in general?
2. Can Euler's formula be used to represent complex waves?
3. What are the applications of combining sine and cosine waves in this way?
4. How does this result relate to phasor representation in physics?
5. Can you derive similar expressions using hyperbolic functions?

**Tip:** When handling trigonometric expressions, Euler’s formula is a powerful tool for converting between sine, cosine, and complex exponentials, simplifying many problems in physics and engineering!

Show using Euler's formula that A cos(kx − ωt) + B sin(kx − ωt) can be equivalently written as C cos(kx − ωt + φ). Find C and φ in terms of A and B.

Learn how to use Euler's formula to express A cos(kx − ωt) + B sin(kx − ωt) as C cos(kx − ωt + φ). This step-by-step solution involves key concepts like trigonometry, complex numbers, and polar coordinates. Suitable for undergraduate-level students.

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