Let's solve the problem step by step. ### Given: **Polar curve:** \( r = 3 \cos\left(\frac{\theta}{2}\right) \) **Interval:** \( 0 \leq \theta \leq 2\pi \) To fill the table, follow these steps: 1. **Calculate \( r \) for each \( \theta \).** 2. **Write the polar coordinate \((r,\theta)\).** 3. **Convert to Cartesian coordinates:** - \( x = r\cos\theta \) - \( y = r\sin\theta \) - The Cartesian coordinate is \((x, y)\). --- Let's complete the first few rows as explicit examples: --- ### 1. \( \theta = 0 \) - \( r = 3 \cos(0/2) = 3 \cos(0) = 3 \times 1 = 3 \) - Polar: \( (3, 0) \) - \( x = 3 \cos(0) = 3 \) - \( y = 3 \sin(0) = 0 \) - Cartesian: \( (3, 0) \) --- ### 2. \( \theta = \frac{\pi}{6} \) - \( r = 3 \cos\left(\frac{\pi}{12}\right) \) - \( \cos(\frac{\pi}{12}) \approx 0.9659 \) - \( r \approx 3 \times 0.9659 = 2.8977 \) - Polar: \( (2.8977, \frac{\pi}{6}) \) - \( x = 2.8977 \cos(\frac{\pi}{6}) \approx 2.8977 \times 0.8660 = 2.507 \) - \( y = 2.8977 \sin(\frac{\pi}{6}) \approx 2.8977 \times 0.5 = 1.449 \) - Cartesian: \( (2.507, 1.449) \) --- ### 3. \( \theta = \frac{\pi}{4} \) - \( r = 3 \cos(\frac{\pi}{8}) \) - \( \cos(\frac{\pi}{8}) \approx 0.9239 \) - \( r \approx 3 \times 0.9239 = 2.7717 \) - Polar: \( (2.7717, \frac{\pi}{4}) \) - \( x = 2.7717 \cos(\frac{\pi}{4}) \approx 2.7717 \times 0.7071 = 1.960 \) - \( y = 2.7717 \sin(\frac{\pi}{4}) \approx 2.7717 \times 0.7071 = 1.960 \) - Cartesian: \( (1.960, 1.960) \) --- Let's do a few more: ### 4. \( \theta = \frac{\pi}{3} \) - \( r = 3 \cos(\frac{\pi}{6}) \) - \( \cos(\frac{\pi}{6}) = 0.8660 \) - \( r = 3 \times 0.8660 = 2.598 \) - Polar: \( (2.598, \frac{\pi}{3}) \) - \( x = 2.598 \cos(\frac{\pi}{3}) = 2.598 \times 0.5 = 1.299 \) - \( y = 2.598 \sin(\frac{\pi}{3}) = 2.598 \times 0.8660 = 2.249 \) - Cartesian: \( (1.299, 2.249) \) --- ### 5. \( \theta = \frac{\pi}{2} \) - \( r = 3 \cos(\frac{\pi}{4}) \) - \( \cos(\frac{\pi}{4}) = 0.7071 \) - \( r = 3 \times 0.7071 = 2.121 \) - Polar: \( (2.121, \frac{\pi}{2}) \) - \( x = 2.121 \cos(\frac{\pi}{2}) = 0 \) - \( y = 2.121 \sin(\frac{\pi}{2}) = 2.121 \) - Cartesian: \( (0, 2.121) \) --- ### 6. \( \theta = \pi \) - \( r = 3 \cos(\frac{\pi}{2}) \) - \( \cos(\frac{\pi}{2}) = 0 \) - \( r = 0 \) - Polar: \( (0, \pi) \) - \( x = 0 \) - \( y = 0 \) - Cartesian: \( (0, 0) \) --- ### 7. \( \theta = 2\pi \) - \( r = 3 \cos(\pi) = 3 \times (-1) = -3 \) - Polar: \( (-3, 2\pi) \) - In Cartesian: - \( x = -3 \cos(2\pi) = -3 \times 1 = -3 \) - \( y = -3 \sin(2\pi) = 0 \) - Cartesian: \( (-3, 0) \) --- You would continue in this way for each table entry, plugging in the values of \( \theta \) to compute \( r \), then \( x \) and \( y \). --- ## Table (First Few Rows) | θ | r = 3cos(θ/2) | (r, θ) | x | y | (x, y) | |-----------|---------------|--------------------------|--------|--------|-------------| | 0 | 3 | (3, 0) | 3 | 0 | (3, 0) | | π/6 | 2.8977 | (2.8977, π/6) | 2.507 | 1.449 | (2.507,1.449)| | π/4 | 2.7717 | (2.7717, π/4) | 1.960 | 1.960 | (1.960,1.960)| | π/3 | 2.598 | (2.598, π/3) | 1.299 | 2.249 | (1.299,2.249)| | π/2 | 2.121 | (2.121, π/2) | 0 | 2.121 | (0,2.121) | | ... | ... | ... | ... | ... | ... | | π | 0 | (0, π) | 0 | 0 | (0,0) | | ... | ... | ... | ... | ... | ... | | 2π | -3 | (-3, 2π) | -3 | 0 | (-3,0) | --- ### (b) Sketching the Graph - The graph of \( r = 3 \cos(\theta/2) \) is a **limacon with an inner loop**. - Plot each (x, y) point calculated above. - You should see the curve start at (3, 0), loop around the origin at θ = π (where r = 0), and end at (-3, 0) for θ = 2π. --- **Let me know if you'd like more rows filled, a plotted graph, or any further explanation!**