Let's break down the problem step by step: --- ### **Step 1: Understanding the Sine Wave** The general form of a sine wave is: \[ V(t) = V_{max} \sin(\theta) \] where \( V_{max} = 50 \) V. --- ### **Step 2: Approximation Method** **(a) Average Value Over Half Cycle** A sine wave from \(0\) to \(\pi\) (half a cycle) starts at 0, reaches 50 V at \(\pi/2\), and returns to 0 at \(\pi\). The average can be approximated as the area under the curve divided by the interval. **Approximate Average:** - The average is roughly the midpoint value for a smooth curve like sine over half a cycle. - At midpoint (\(\theta = \pi/2\)), \( V = 50 \sin(\pi/2) = 50 \) V. - The value starts at 0, rises to 50, and returns to 0, so the average is about \( \frac{2}{\pi} \times 50 \approx 31.8 \) V (this is a known result, but let's check via calculus below). **(b) Average Value Over Full Cycle** For a full cycle (\(0\) to \(2\pi\)), the positive half and negative half cancel out, so the average **over a full cycle** is **0 V**. --- ### **Step 3: Calculus Integration for Exact Average** #### **(a) Half Cycle:** \[ \text{Average} = \frac{1}{\pi} \int_0^\pi 50 \sin(\theta) \, d\theta \] Calculate the integral: \[ \int_0^\pi \sin(\theta) d\theta = [-\cos(\theta)]_0^\pi = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2 \] So, \[ \text{Average} = \frac{1}{\pi} \cdot 50 \cdot 2 = \frac{100}{\pi} \approx 31.83 \text{ V} \] #### **(b) Full Cycle:** \[ \text{Average} = \frac{1}{2\pi} \int_0^{2\pi} 50 \sin(\theta) d\theta \] \[ \int_0^{2\pi} \sin(\theta) d\theta = [-\cos(\theta)]_0^{2\pi} = -\cos(2\pi) + \cos(0) = -1 + 1 = 0 \] So, \[ \text{Average} = 0 \] --- ### **Step 4: Final Answers** **(a) Average value for half a cycle (\(0\) to \(\pi\)):** \[ \boxed{31.83~\text{V}} \] **(b) Average value for a complete cycle (\(0\) to \(2\pi\)):** \[ \boxed{0~\text{V}} \] --- **Explanation:** - The average over half a cycle is about 63.7% of the peak value (for a sine wave). - Over a full cycle, positive and negative halves cancel out, so average = 0. Let me know if you want the graph or further explanation!