a. Copy and complete the polloring table of values for \( y=3 \sin \theta+\cos \theta \) busing a scale of 2 cm to \( 30^{\circ} \) on the \( \theta \) caxis and 2 cm to 1 unit on they \( y \)-axis, draw the graph of \( y=3 \sin \theta+\cos \theta \) कor \( \theta^{\circ} \leq \theta \leq 160^{\circ} \). c. Use your graph to find if the solution set 0 . the equation \( \cos \sin ^{\theta} \theta+2 \cos \theta-1=0 \) uf the maximum value of \( y \) correct to ond dexmal plo \( \sin \)

Let's break down the problem step by step. ### Part a: Completing the Table of Values We need to evaluate the function \( y = 3 \sin \theta + \cos \theta \) for various values of \( \theta \) in the range \( 0^\circ \) to \( 160^\circ \). We will choose increments of \( 30^\circ \) for our calculations. 1. **Calculate \( y \) for each \( \theta \)**: - \( \theta = 0^\circ \) - \( \theta = 30^\circ \) - \( \theta = 60^\circ \) - \( \theta = 90^\circ \) - \( \theta = 120^\circ \) - \( \theta = 150^\circ \) - \( \theta = 160^\circ \) Let's calculate the values of \( y \) for these angles. ### Calculating Values 1. **For \( \theta = 0^\circ \)**: \[ y = 3 \sin(0) + \cos(0) = 0 + 1 = 1 \] 2. **For \( \theta = 30^\circ \)**: \[ y = 3 \sin(30) + \cos(30) = 3 \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{3}{2} + \frac{\sqrt{3}}{2} \approx 2.366 \] 3. **For \( \theta = 60^\circ \)**: \[ y = 3 \sin(60) + \cos(60) = 3 \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{3\sqrt{3}}{2} + \frac{1}{2} \approx 3.598 \] 4. **For \( \theta = 90^\circ \)**: \[ y = 3 \sin(90) + \cos(90) = 3 \cdot 1 + 0 = 3 \] 5. **For \( \theta = 120^\circ \)**: \[ y = 3 \sin(120) + \cos(120) = 3 \cdot \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{3\sqrt{3}}{2} - \frac{1}{2} \approx 2.598 \] 6. **For \( \theta = 150^\circ \)**: \[ y = 3 \sin(150) + \cos(150) = 3 \cdot \frac{1}{2} - \frac{\sqrt{3}}{2} = \frac{3}{2} - \frac{\sqrt{3}}{2} \approx 0.634 \] 7. **For \( \theta = 160^\circ \)**: \[ y = 3 \sin(160) + \cos(160) = 3 \cdot \sin(160) + \cos(160) \approx 3 \cdot 0.9397 - 0.9397 \approx 2.819 \] ### Table of Values | \( \theta \) (degrees) | \( y \) | |------------------------|-----------------------| | 0 | 1 | | 30 | 2.366 | | 60 | 3.598 | | 90 | 3 | | 120 | 2.598 | | 150 | 0.634 | | 160 | 2.819 | ### Part b: Drawing the Graph To draw the graph of \( y = 3 \sin \theta + \cos \theta \) using the table of values: - Use a scale of 2 cm for \( 30^\circ \) on the \( \theta \) axis. - Use a scale of 2 cm for 1 unit on the \( y \) axis. Plot the points from the table and connect them smoothly to form the graph. ### Part c: Finding the Maximum Value of \( y \) From the table, the maximum value of \( y \) occurs at \( \theta = 60^\circ \) where \( y \approx 3.598 \). Now, let's solve the equation \( \cos(\sin \theta) + 2 \cos \theta - 1 = 0 \) to find the solution set. ### Solving the Equation We can rewrite the equation as: \[ \cos(\sin \theta) + 2 \cos \theta - 1 = 0 \] This is a transcendental equation and may require numerical methods or graphing to find solutions. However, we can analyze the maximum value of \( y \) which we found to be approximately \( 3.6 \). ### Conclusion - The maximum value of \( y \) is approximately \( 3.6 \) (to one decimal place). - The solution set for the equation \( \cos(\sin \theta) + 2 \cos \theta - 1 = 0 \) can be found graphically or numerically, but it is not directly related to the maximum value of \( y \). If you need further assistance with the equation or any other part, please let me know!