Let's solve the given equation [tex]\(2 \tan(\theta) + 2 = 0\)[/tex] step-by-step:
### Part (a): Solve for all degree solutions.
1. Rearrange the equation:
[tex]\[
2 \tan(\theta) + 2 = 0
\][/tex]
Subtract 2 from both sides:
[tex]\[
2 \tan(\theta) = -2
\][/tex]
Divide both sides by 2:
[tex]\[
\tan(\theta) = -1
\][/tex]
2. Find the general solution for [tex]\(\theta\)[/tex]:
The tangent function has a period of [tex]\(180^\circ\)[/tex]. Therefore, the general solution for [tex]\(\tan(\theta) = -1\)[/tex] can be written as:
[tex]\[
\theta = 135^\circ + 180^\circ k
\][/tex]
where [tex]\(k\)[/tex] is any integer.
So, the general solution is:
[tex]\[
\boxed{\theta = 135^\circ + 180^\circ k}
\][/tex]
### Part (b): Solve for [tex]\(\theta\)[/tex] in the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex].
1. Find specific solutions within the interval:
We need to find the values of [tex]\(\theta\)[/tex] that satisfy the equation [tex]\( \tan(\theta) = -1 \)[/tex] within the range [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex].
2. Substitute [tex]\(k\)[/tex] into the general solution:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
\theta = 135^\circ
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
\theta = 135^\circ + 180^\circ = 315^\circ
\][/tex]
- For [tex]\(k = -1\)[/tex]:
[tex]\[
\theta = 135^\circ - 180^\circ = -45^\circ
\][/tex]
Since [tex]\(-45^\circ\)[/tex] is not in the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex], we discard this value.
3. Verify all unique solutions within the given range:
We have:
[tex]\[
135^\circ, 315^\circ
\][/tex]
So, the solutions for [tex]\(\theta\)[/tex] in the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex] are:
[tex]\[
\boxed{135^\circ, 315^\circ}
\][/tex]
To summarize:
(a) The general solution for the equation [tex]\(2 \tan(\theta) + 2 = 0\)[/tex] is:
[tex]\[
\theta = 135^\circ + 180^\circ k
\][/tex]
(b) The specific solutions for [tex]\(\theta\)[/tex] within the range [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex] are:
[tex]\[
135^\circ, 315^\circ
\][/tex]
