To find the value of [tex]\(\theta\)[/tex] for which [tex]\(\tan \theta\)[/tex] is undefined in the interval [tex]\(180^\circ \leq \theta < 360^\circ\)[/tex], we need to understand under what conditions the tangent function is undefined. The tangent of an angle [tex]\(\theta\)[/tex] is given by:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
For [tex]\(\tan \theta\)[/tex] to be undefined, [tex]\(\cos \theta\)[/tex] must be equal to zero, because division by zero is undefined.
Next, let's identify the angles within the given interval where [tex]\(\cos \theta = 0\)[/tex]. The cosine function is zero at angles of [tex]\(90^\circ\)[/tex] plus multiples of [tex]\(180^\circ\)[/tex]:
[tex]\[
\cos \theta = 0 \implies \theta = 90^\circ + 180^\circ \cdot n
\][/tex]
where [tex]\(n\)[/tex] is an integer.
Within the interval [tex]\(180^\circ \leq \theta < 360^\circ\)[/tex], we need to find the appropriate integer value of [tex]\(n\)[/tex] that satisfies this condition. There are two relevant multiples to consider in this interval:
1. When [tex]\(n = 1\)[/tex],
[tex]\[
\theta = 90^\circ + 180^\circ \cdot 1 = 90^\circ + 180^\circ = 270^\circ
\][/tex]
Thus, [tex]\(\theta = 270^\circ\)[/tex] lies within the specified interval [tex]\(180^\circ \leq \theta < 360^\circ\)[/tex]. Checking the other potential values, for example,
2. When [tex]\(n = 2\)[/tex],
[tex]\[
\theta = 90^\circ + 180^\circ \cdot 2 = 90^\circ + 360^\circ = 450^\circ
\][/tex]
However, 450^\circ is outside the given interval.
So, the only valid angle in the range [tex]\(180^\circ \leq \theta < 360^\circ\)[/tex] where [tex]\(\cos \theta = 0\)[/tex], and hence [tex]\(\tan \theta\)[/tex] is undefined, is:
[tex]\[
\theta = 270^\circ
\][/tex]
Therefore, the measure of [tex]\(\theta\)[/tex] is 270 degrees.
