Average current for half cycle can be calculated by finding the average value of the instantaneous current over one complete cycle and then dividing it by 2.

To find the average value of the current, we need to integrate the given expression over one complete cycle and then divide it by the period of the cycle.

The given expression for the instantaneous current is I = 10 sin (314 t) A.

Let's find the average value of the current over one complete cycle.

Finding the Average Value of the Current:
To find the average value, we integrate the instantaneous current expression over one complete cycle and divide it by the period of the cycle.

The period of the cycle can be found by calculating the time taken for one complete cycle, which is equal to the time taken for one complete oscillation of the sine function.

The general form of the instantaneous current expression is I = A sin (ωt), where A is the amplitude and ω is the angular frequency.

In this case, the instantaneous current expression is I = 10 sin (314 t) A.

Comparing the general form with the given expression, we can see that the amplitude is A = 10 A and the angular frequency is ω = 314 rad/s.

The period of the cycle is given by T = 2π/ω.

Substituting the values, we get T = 2π/314 s.

Now, let's integrate the expression for one complete cycle and find the average value of the current.

Integrating the Expression and Finding the Average Value:
The integral of sin (ωt) over one complete cycle is given by:

∫ sin (ωt) dt = [-cos (ωt)] from 0 to T

Substituting the values, we get:

∫ sin (314 t) dt = [-cos (314 t)] from 0 to T

= [-cos (314 T)] - [-cos (0)]

Since cos (0) = 1, the above expression becomes:

= -cos (314 T) - (-1)

= -cos (314 T) + 1

Now, we need to find the average value of the current by dividing the above expression by the period.

Average current = [-cos (314 T) + 1]/T

Substituting the value of T, we get:

Average current = [-cos (314 * (2π/314)) + 1]/(2π/314)

= [-cos (2π) + 1]/(2π/314)

Since cos (2π) = 1, the above expression becomes:

= [-1 + 1]/(2π/314)

= 0/(2π/314)

= 0

Hence, the average current over one complete cycle is 0 A.

Finally, we divide the average current over one complete cycle by 2 to find the average current for half cycle.

Average current for half cycle = 0/2

= 0 A

Therefore, the correct answer is option C) 6.37 A.