To derive the angle sum formulas for cosine and sine using Euler's Theorem, we start with the given equation:

$e^{i\theta} e^{i\theta'} = e^{i(\theta + \theta')}.$

Step 1: Expand both sides using Euler's formula

Euler's formula states that $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. Substituting this for both $e^{i\theta}$ and $e^{i\theta'}$, we have:

$(\cos(\theta) + i\sin(\theta))(\cos(\theta') + i\sin(\theta')) = \cos(\theta + \theta') + i\sin(\theta + \theta').$

Step 2: Expand the left-hand side

Using the distributive property, expand $(\cos(\theta) + i\sin(\theta))(\cos(\theta') + i\sin(\theta'))$:

$= \cos(\theta)\cos(\theta') + i\cos(\theta)\sin(\theta') + i\sin(\theta)\cos(\theta') + i^2\sin(\theta)\sin(\theta').$

Since $i^2 = -1$, simplify:

$= \cos(\theta)\cos(\theta') - \sin(\theta)\sin(\theta') + i(\cos(\theta)\sin(\theta') + \sin(\theta)\cos(\theta')).$

Step 3: Equate real and imaginary parts

On the right-hand side, $\cos(\theta + \theta') + i\sin(\theta + \theta')$, the real part corresponds to $\cos(\theta + \theta')$ and the imaginary part corresponds to $\sin(\theta + \theta')$. Equating the real and imaginary parts of both sides:

1. Real part: $\cos(\theta + \theta') = \cos(\theta)\cos(\theta') - \sin(\theta)\sin(\theta').$

2. Imaginary part: $\sin(\theta + \theta') = \cos(\theta)\sin(\theta') + \sin(\theta)\cos(\theta').$

Final Result: Angle Sum Formulas

From the derivation, we obtain:

$\cos(\theta + \theta') = \cos(\theta)\cos(\theta') - \sin(\theta)\sin(\theta'),$ $\sin(\theta + \theta') = \cos(\theta)\sin(\theta') + \sin(\theta)\cos(\theta').$

These are the angle sum formulas for cosine and sine.

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Do you have any questions or would you like more details?

Here are 5 related questions to explore:

1. How does Euler's formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ connect complex numbers and trigonometry?
2. Can we derive the angle difference formulas for cosine and sine similarly?
3. How do these angle sum formulas apply in solving trigonometric identities?
4. Why does the imaginary unit $i^2 = -1$ play a key role in this derivation?
5. What are some practical applications of the angle sum formulas in physics or engineering?

Tip: Always use Euler's formula as a powerful tool to simplify and derive trigonometric identities efficiently!