If an alternating current is given by i = a sin (`omega`t) + b cos (`omega`t), then RMS value of current is :-

Options

A. `a/sqrt2+b/sqrt2`
B. `(a+b)/sqrt2`
C. `sqrt((a^2+b^2)/2)`
D. `(a^2+b^2)/sqrt2`

Explanation

To find the RMS (Root Mean Square) value of the alternating current given by the equation \( i = a \sin(\omega t) + b \cos(\omega t) \), we can follow these steps:

### Step 1: Identify the form of the current
The current is expressed as:
\[ i = a \sin(\omega t) + b \cos(\omega t) \]
where \( a \) and \( b \) are constants representing the amplitudes of the sine and cosine components, respectively.

### Step 2: Determine the peak value of the current
The peak value (maximum value) of a function of the form \( A \sin(\theta) + B \cos(\theta) \) can be calculated using the formula:
\[ \text{Peak value} = \sqrt{A^2 + B^2} \]
In our case, \( A = a \) and \( B = b \). Therefore, the peak value \( I_{\text{peak}} \) is:
\[ I_{\text{peak}} = \sqrt{a^2 + b^2} \]

### Step 3: Calculate the RMS value
The RMS value of an alternating current is given by the formula:
\[ I_{\text{RMS}} = \frac{I_{\text{peak}}}{\sqrt{2}} \]
Substituting the peak value we found in Step 2:
\[ I_{\text{RMS}} = \frac{\sqrt{a^2 + b^2}}{\sqrt{2}} = \frac{\sqrt{a^2 + b^2}}{2} \]

### Step 4: Simplify the expression
We can simplify the expression for the RMS value:
\[ I_{\text{RMS}} = \frac{1}{\sqrt{2}} \sqrt{a^2 + b^2} \]
This can also be expressed as:
\[ I_{\text{RMS}} = \frac{a^2 + b^2}{2} \]

### Final Result
Thus, the RMS value of the current is:
\[ I_{\text{RMS}} = \frac{\sqrt{a^2 + b^2}}{\sqrt{2}} \]