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

Options

A. `sqrt(2)A`
B. `2sqrt(2+sqrt(3))`
C. 4
D. `2sqrt(2)A`

Explanation

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

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

we will follow these steps:

### Step 1: Rewrite the cosine term
We can rewrite the cosine term using the sine function. Recall that:

\[ \cos(x) = \sin\left(\frac{\pi}{2} - x\right) \]

Thus, we can express \( \cos(100 \pi t + 30^\circ) \) as:

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

### Step 2: Combine the sine terms
Now, we can rewrite the current \( i(t) \):

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

### Step 3: Use the sine addition formula
We can use the sine addition formula:

\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

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

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

### Step 4: Apply the formula
Now, applying the sine addition formula:

\[ i(t) = 2 \cdot 2 \sin\left(\frac{(100 \pi t) + (60^\circ - 100 \pi t)}{2}\right) \cos\left(\frac{(100 \pi t) - (60^\circ - 100 \pi t)}{2}\right) \]

This simplifies to:

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

### Step 5: Calculate the effective (RMS) value
The effective value (RMS value) of a sinusoidal current is given by:

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

Where \( I_0 \) is the amplitude of the current. In our case, the amplitude \( I_0 \) is \( 4 \sin(30^\circ) \).

Since \( \sin(30^\circ) = \frac{1}{2} \):

\[ I_0 = 4 \cdot \frac{1}{2} = 2 \, \text{A} \]

Thus,

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

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

\[ \boxed{\sqrt{2} \, \text{A}} \]