An AC current is given by` i(t) = I_(1)sin wt + I_(2)cos wt`. Find the rms value of current.

Answer

To find the RMS (Root Mean Square) value of the given AC current \( i(t) = I_1 \sin(\omega t) + I_2 \cos(\omega t) \), we will follow these steps:

### Step 1: Rewrite the cosine term
We can express the cosine term in terms of sine to combine both terms. Recall that:
\[ \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right) \]
Thus, we can rewrite the current as:
\[ i(t) = I_1 \sin(\omega t) + I_2 \sin\left(\omega t + \frac{\pi}{2}\right) \]

### Step 2: Identify the phase difference
The two terms \( I_1 \sin(\omega t) \) and \( I_2 \sin\left(\omega t + \frac{\pi}{2}\right) \) have a phase difference of \( \frac{\pi}{2} \) radians (90 degrees).

### Step 3: Calculate the resultant amplitude
The resultant amplitude \( I \) of two perpendicular components can be calculated using the Pythagorean theorem:
\[ I = \sqrt{I_1^2 + I_2^2} \]

### Step 4: Find the RMS value
The RMS value of a sinusoidal current is given by:
\[ I_{\text{rms}} = \frac{I}{\sqrt{2}} \]
Substituting the expression for \( I \):
\[ I_{\text{rms}} = \frac{1}{\sqrt{2}} \sqrt{I_1^2 + I_2^2} \]

### Final Expression
Thus, the RMS value of the current \( i(t) \) is:
\[ I_{\text{rms}} = \frac{1}{\sqrt{2}} \sqrt{I_1^2 + I_2^2} \]