An alternating current is given by
`I = i_1 cos omegat + i_2 sin omegat`.
The rms current is given by

Options

A. `(i_1 + i_2)/(sqrt2)`
B. `(|i_1 + i_2|)/(sqrt2)`
C. `sqrt((i_1^2 + i_2^2)/(2)`
D. `sqrt((i_1^2 + i_2^2)/(sqrt2)`

Answer

To find the RMS (Root Mean Square) current for the given alternating current \( I = i_1 \cos(\omega t) + i_2 \sin(\omega t) \), we can follow these steps:

### Step 1: Identify the components of the current
The given current can be expressed as:
\[ I = i_1 \cos(\omega t) + i_2 \sin(\omega t) \]
Here, \( i_1 \) and \( i_2 \) are the amplitudes of the cosine and sine components, respectively.

### Step 2: Use the trigonometric identity
To combine the two components, we can express the cosine function in terms of sine:
\[ \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right) \]
Thus, we can rewrite the equation as:
\[ I = i_1 \sin\left(\omega t + \frac{\pi}{2}\right) + i_2 \sin(\omega t) \]

### Step 3: Combine the sine terms
Using the sine addition formula, we can combine these terms into a single sine function:
\[ I = \sqrt{i_1^2 + i_2^2} \sin\left(\omega t + \phi\right) \]
where \( \phi \) is the phase difference between the two components.

### Step 4: Calculate the RMS value
The RMS value of a sinusoidal current is given by:
\[ I_{\text{RMS}} = \frac{I_{\text{max}}}{\sqrt{2}} \]
where \( I_{\text{max}} \) is the maximum (peak) current. In our case, the maximum current is:
\[ I_{\text{max}} = \sqrt{i_1^2 + i_2^2} \]
Thus, substituting this into the RMS formula gives:
\[ I_{\text{RMS}} = \frac{\sqrt{i_1^2 + i_2^2}}{\sqrt{2}} = \frac{\sqrt{i_1^2 + i_2^2}}{2} \]

### Final Result
Therefore, the RMS current is:
\[ I_{\text{RMS}} = \frac{\sqrt{i_1^2 + i_2^2}}{\sqrt{2}} \]