Find all solutions to each equation. Your answer will involve " \( n \) ". a) \( \tan \theta=-1 \)

The equation \( \tan \theta = -1 \) means that the angle \( \theta \) can be found where the tangent function takes the value of -1. This occurs at \( \theta = -\frac{\pi}{4} + n\pi \), where \( n \) is any integer. The reason for adding \( n\pi \) is because the tangent function has a period of \( \pi \), meaning it repeats every \( \pi \) radians. So, all solutions can be expressed as: \[ \theta = -\frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \] Just think of it as swinging through a cycle of angles; you meet the -1 customer every half-turn!