To solve for all angles [tex]\(\theta\)[/tex] in the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex] that satisfy the equation
[tex]\[
\tan(\theta) = -\frac{1}{2},
\][/tex]
we need to determine the angles whose tangent value is [tex]\(-\frac{1}{2}\)[/tex]. Let's go through the steps:
1. Find the Principal Angle:
The first step is to find the principal angle whose tangent is [tex]\(-\frac{1}{2}\)[/tex]. Use the inverse tangent function to find this angle. We know that:
[tex]\[
\theta = \tan^{-1}\left(-\frac{1}{2}\right).
\][/tex]
In degrees, the closest value is approximately [tex]\(-26.6^\circ\)[/tex].
2. Normalize the Principal Angle:
Since [tex]\(-26.6^\circ\)[/tex] is not within the interval [tex]\([0^\circ, 360^\circ)\)[/tex], we need to normalize it by adding 360 degrees to it:
[tex]\[
\theta_1 = -26.6^\circ + 360^\circ = 333.4^\circ.
\][/tex]
3. Determine the Periodicity:
The tangent function is periodic with a period of 180 degrees. This means that any angle [tex]\(\theta\)[/tex] can be expressed as:
[tex]\[
\theta = \theta_1 + k \cdot 180^\circ,
\][/tex]
where [tex]\(k\)[/tex] is an integer.
4. Find Other Angles in the Range:
Starting with the principal angle [tex]\(\theta_1 = 333.4^\circ\)[/tex], we add 180 degrees to find the next angle:
[tex]\[
\theta_2 = 333.4^\circ + 180^\circ = 513.4^\circ.
\][/tex]
However, [tex]\(513.4^\circ\)[/tex] is outside the interval [tex]\([0^\circ, 360^\circ)\)[/tex], so it is not a valid solution within the given range.
5. Listing the Valid Angles:
Therefore, the only solution within the interval [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex] is:
[tex]\[
\theta = 333.4^\circ.
\][/tex]
Thus, the angle that satisfies [tex]\(\tan(\theta) = -\frac{1}{2}\)[/tex] in the specified range is:
[tex]\[
\boxed{333.4^\circ}
\][/tex]
