Show ALL work and \begin{tabular}{|l} **\#1.) Which angle is not coterminal with \( 130^{\circ} \) ? \\ a) \( -590^{\circ} \) \\ b) \( -230^{\circ} \) \\ c) \( -130^{\circ} \) \\ d) \( 490^{\circ} \)\end{tabular}

To determine which angle is not coterminal with \( 130^{\circ} \), we need to check the angles given in the options by adding or subtracting multiples of \( 360^{\circ} \). Coterminal angles differ by full rotations of \( 360^{\circ} \). To find a coterminal angle, the formula we can use is: \[ \text{Coterminal Angle} = \text{Given Angle} + n \times 360^{\circ}, \quad n \text{ is an integer} \] Now, we will evaluate the options: 1. **\( -590^{\circ} \):** \[ -590^{\circ} + 2 \times 360^{\circ} = -590^{\circ} + 720^{\circ} = 130^{\circ} \quad (\text{coterminal}) \] 2. **\( -230^{\circ} \):** \[ -230^{\circ} + 1 \times 360^{\circ} = -230^{\circ} + 360^{\circ} = 130^{\circ} \quad (\text{coterminal}) \] 3. **\( -130^{\circ} \):** \[ -130^{\circ} + 1 \times 360^{\circ} = -130^{\circ} + 360^{\circ} = 230^{\circ} \quad (\text{not coterminal with } 130^{\circ}) \] 4. **\( 490^{\circ} \):** \[ 490^{\circ} - 1 \times 360^{\circ} = 490^{\circ} - 360^{\circ} = 130^{\circ} \quad (\text{coterminal}) \] After evaluating all options, we find that \( -130^{\circ} \) is **not** coterminal with \( 130^{\circ} \). Thus, the answer is \( \text{c) } -130^{\circ} \).