Find the effective value of current.
`i=2 sin 100 (pi)t + 2 cos (100 pi t + 30^(@))`.

Explanation

To find the effective value of the current given by the equation:

\[ i = 2 \sin(100 \pi t) + 2 \cos(100 \pi t + 30^\circ) \]

we will follow these steps:

### Step 1: Rewrite the cosine term as a sine term
Using the trigonometric identity \( \cos \theta = \sin(90^\circ - \theta) \), we can rewrite the cosine term:

\[ \cos(100 \pi t + 30^\circ) = \sin\left(90^\circ - (100 \pi t + 30^\circ)\right) = \sin(60^\circ - 100 \pi t) \]

### Step 2: Substitute back into the equation
Now we can rewrite the current \( i \):

\[ i = 2 \sin(100 \pi t) + 2 \sin(60^\circ - 100 \pi t) \]

### Step 3: Use the sine addition formula
We can use the sine addition formula \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \) to combine the two sine terms:

Let \( A = 100 \pi t \) and \( B = 60^\circ - 100 \pi t \).

Calculating \( \frac{A + B}{2} \) and \( \frac{A - B}{2} \):

\[ \frac{A + B}{2} = \frac{100 \pi t + (60^\circ - 100 \pi t)}{2} = \frac{60^\circ}{2} = 30^\circ \]

\[ \frac{A - B}{2} = \frac{100 \pi t - (60^\circ - 100 \pi t)}{2} = \frac{200 \pi t - 60^\circ}{2} = 100 \pi t - 30^\circ \]

Thus, we can write:

\[ i = 4 \sin(30^\circ) \cos(100 \pi t - 30^\circ) \]

### Step 4: Calculate \( \sin(30^\circ) \)
We know that \( \sin(30^\circ) = \frac{1}{2} \):

\[ i = 4 \cdot \frac{1}{2} \cos(100 \pi t - 30^\circ) = 2 \cos(100 \pi t - 30^\circ) \]

### Step 5: Identify the amplitude
From the equation \( i = 2 \cos(100 \pi t - 30^\circ) \), we can see that the amplitude \( I_0 \) is 2 A.

### Step 6: Calculate the effective (RMS) value of the current
The effective value (RMS) of the current is given by:

\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \text{ A} \]

### Final Answer
The effective value of the current is:

\[ I_{\text{rms}} = \sqrt{2} \text{ A} \]