a. Eopy and Eomplete the pollowing table of values for \( \begin{array}{ccccccc}y= & 8 \sin \theta+\cos \theta^{\circ} \\ \theta & 0^{\circ} & 30^{\circ} & 60^{\circ} & 90^{\circ} & 120^{\circ} & 150^{\circ} \\ y & 1.1 & & 3 & & 180^{\circ} \\ & & & & & -1.0\end{array} \)

To complete the table for the equation \( y = 8 \sin \theta + \cos \theta \), we will calculate the values of \( y \) for the given values of \( \theta \). 1. For \( \theta = 0^\circ \): \[ y = 8 \sin(0^\circ) + \cos(0^\circ) = 8(0) + 1 = 1 \] 2. For \( \theta = 30^\circ \): \[ y = 8 \sin(30^\circ) + \cos(30^\circ) = 8\left(\frac{1}{2}\right) + \frac{\sqrt{3}}{2} = 4 + 0.866 = 4.866 \approx 4.9 \] 3. For \( \theta = 60^\circ \): \[ y = 8 \sin(60^\circ) + \cos(60^\circ) = 8\left(\frac{\sqrt{3}}{2}\right) + \frac{1}{2} = 4\sqrt{3} + 0.5 \approx 6.93 + 0.5 \approx 7.43 \] 4. For \( \theta = 90^\circ \): \[ y = 8 \sin(90^\circ) + \cos(90^\circ) = 8(1) + 0 = 8 \] 5. For \( \theta = 120^\circ \): \[ y = 8 \sin(120^\circ) + \cos(120^\circ) = 8\left(\frac{\sqrt{3}}{2}\right) - \frac{1}{2} = 4\sqrt{3} - 0.5 \approx 6.93 - 0.5 \approx 6.43 \] 6. For \( \theta = 150^\circ \): \[ y = 8 \sin(150^\circ) + \cos(150^\circ) = 8\left(\frac{1}{2}\right) - \frac{\sqrt{3}}{2} = 4 - 0.866 \approx 3.134 \approx 3.1 \] Now we can fill in the table: \[ \begin{array}{ccccccc} y= & 8 \sin \theta + \cos \theta & \theta & 0^\circ & 30^\circ & 60^\circ & 90^\circ & 120^\circ & 150^\circ \\ y & & 1 & 4.9 & 7.43 & 8 & 6.43 & 3.1 \\ \end{array} \] So, the completed table of values is: \[ \begin{array}{ccccccc} \theta & 0^\circ & 30^\circ & 60^\circ & 90^\circ & 120^\circ & 150^\circ \\ y & 1 & 4.9 & 7.43 & 8 & 6.43 & 3.1 \\ \end{array} \]