Let's analyze and organize the information for the polar equation: **r = 3 + 5cos(θ)** ### Step 1: Create a Table of Values We'll pick key values of θ (in radians and degrees) and compute \( r \): | θ (radians) | θ (degrees) | cos(θ) | r = 3 + 5cos(θ) | |:-----------:|:-----------:|:------:|:---------------:| | 0 | 0° | 1 | 3 + 5×1 = 8 | | π/3 | 60° | 0.5 | 3 + 5×0.5 = 5.5| | π/2 | 90° | 0 | 3 + 5×0 = 3 | | 2π/3 | 120° | -0.5 | 3 + 5×(-0.5)=0.5| | π | 180° | -1 | 3 + 5×(-1)=-2 | | 4π/3 | 240° | -0.5 | 3 + 5×(-0.5)=0.5| | 3π/2 | 270° | 0 | 3 + 5×0 = 3 | | 5π/3 | 300° | 0.5 | 3 + 5×0.5 = 5.5| | 2π | 360° | 1 | 3 + 5×1 = 8 | ### Step 2: Where does the graph cross r = 0? Set \( r = 0 \): \[ 0 = 3 + 5\cos(\theta) \implies \cos(\theta) = -\frac{3}{5} \] So, \[ \theta = \arccos\left(-\frac{3}{5}\right) \approx 126.87^\circ \text{ and } 233.13^\circ \] or \[ \theta \approx 2.214 \text{ radians and } 4.069 \text{ radians} \] --- ### Step 3: Table Highlighting Key Points Here is the completed table, including where \( r = 0 \): | θ (radians) | θ (degrees) | cos(θ) | r = 3 + 5cos(θ) | |:-----------:|:-----------:|:------:|:---------------:| | 0 | 0° | 1 | 8 | | π/3 | 60° | 0.5 | 5.5 | | π/2 | 90° | 0 | 3 | | 2π/3 | 120° | -0.5 | 0.5 | | **2.214** | **126.87°** | -0.6 | **0** | | π | 180° | -1 | -2 | | **4.069** | **233.13°** | -0.6 | **0** | | 4π/3 | 240° | -0.5 | 0.5 | | 3π/2 | 270° | 0 | 3 | | 5π/3 | 300° | 0.5 | 5.5 | | 2π | 360° | 1 | 8 | --- ### Step 4: Notes for Graphing - The **maximum value of r** is 8 (when θ = 0, 2π). - The **minimum value of r** is -2 (when θ = π). - The curve crosses the pole (r = 0) at approximately θ ≈ 2.214 radians (126.87°) and θ ≈ 4.069 radians (233.13°). --- ### **Summary Table (for your graph):** ``` | θ (degrees) | r | |-------------|-------------| | 0 | 8 | | 60 | 5.5 | | 90 | 3 | | 120 | 0.5 | | 126.87 | 0 | <- crosses pole | 180 | -2 | <- minimum r | 233.13 | 0 | <- crosses pole | 240 | 0.5 | | 270 | 3 | | 300 | 5.5 | | 360 | 8 | ``` Use this table to plot key points and sketch the **limacon** curve for \( r = 3 + 5\cos(\theta) \).