Write the complex exponential function $e^{i\theta }$ as a sum of its real and imaginary parts:

• A. $\sin \left(\theta \right)+i\cos \left(\theta \right)$
• B. $\sin \left(\theta \right)-i\cos \left(\theta \right)$
• C. $\cos \left(\theta \right)+i\sin \left(\theta \right)$
• D. $\cos \left(\theta \right)-i\sin \left(\theta \right)$

Answer: (C)

Verified step by step guidance

1. Recall Euler's formula, which states that the complex exponential function \(e^{i\theta}\) can be expressed as a combination of cosine and sine functions: \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\).
2. Understand that in this expression, \(\cos(\theta)\) represents the real part of the complex number, and \(\sin(\theta)\) multiplied by \(i\) represents the imaginary part.
3. Note that the other options given, such as \(\sin(\theta) + i\cos(\theta)\) or \(\sin(\theta) - i\cos(\theta)\), do not match Euler's formula and thus are not correct representations of \(e^{i\theta}\).
4. To write \(e^{i\theta}\) as a sum of its real and imaginary parts, simply identify the real part as \(\cos(\theta)\) and the imaginary part as \(\sin(\theta)\) multiplied by \(i\), resulting in \(\cos(\theta) + i\sin(\theta)\).
5. This form is fundamental in trigonometry and complex analysis because it connects exponential functions with trigonometric functions, allowing for easier manipulation and understanding of complex numbers.