Answer
The solution set is $$\{135^\circ+180^\circ n, n\in Z\}$$
Work Step by Step
$$\tan\theta+1=0$$
1) Solve the equation over the interval $[0^\circ,360^\circ)$
$$\tan\theta+1=0$$
$$\tan\theta=-1$$
Over the interval $[0^\circ, 360^\circ)$, there are two values of $\theta$ where $\tan\theta=-1$, which are $135^\circ$ and $315^\circ$.
Therefore, $$\theta=\{135^\circ, 315^\circ\}$$
2) Solve the equation for all solutions
To find all solutions, we add the integer multiples of the period of tangent function, which is $180^\circ$, to each solution found in step 1.
- There are two solutions found in step 1, $\{135^\circ, 315^\circ\}$, so it will be written as $\theta=135^\circ+180^\circ n$ and $\theta=315^\circ+180^\circ n$ where $n\in Z$.
- However, a closer look reveals that $135^\circ+180^\circ n$ in fact covers all the possible values of $315^\circ+180^\circ n$, and vice versa. In other words, they are technically the same.
So the solution set can be shortened to $$\{135^\circ+180^\circ n, n\in Z\}$$
