### Step-by-Step Solution to Draw [tex]\( f(\theta) = \tan(\theta) \)[/tex] on the Interval [tex]\([-π, π]\)[/tex]
1. Understand the Function:
- The function [tex]\( f(\theta) = \tan(\theta) \)[/tex] represents the tangent of [tex]\(\theta\)[/tex].
- Tangent is defined as the ratio of the sine and cosine of [tex]\(\theta\)[/tex]: [tex]\( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)[/tex].
- The tangent function has vertical asymptotes where the cosine of [tex]\(\theta\)[/tex] is zero (i.e., where [tex]\(\theta = \frac{\pi}{2} + k\pi\)[/tex] for any integer [tex]\(k\)[/tex]).
2. Identify Critical Points and Asymptotes:
- Within the interval [tex]\([-π, π]\)[/tex], the critical points of the tangent function where it is undefined occur at [tex]\(-\frac{\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{2}\)[/tex], where the cosine function is zero.
3. Graphing the Function:
- The function [tex]\( \tan(\theta) \)[/tex] will have vertical asymptotes at [tex]\(\theta = \pm \frac{\pi}{2} \)[/tex].
- Between the asymptotes at [tex]\(-\frac{\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{2}\)[/tex], the tangent function will smoothly increase from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex].
4. Plot Key Points:
- At [tex]\(\theta = 0\)[/tex], [tex]\( \tan(0) = 0 \)[/tex].
- As [tex]\(\theta\)[/tex] approaches [tex]\(-\frac{\pi}{2}\)[/tex], [tex]\( \tan(\theta) \)[/tex] approaches [tex]\(-\infty\)[/tex].
- As [tex]\(\theta\)[/tex] approaches [tex]\(\frac{\pi}{2}\)[/tex], [tex]\( \tan(\theta) \)[/tex] approaches [tex]\(+\infty\)[/tex].
5. Sketch the Graph:
- Draw the [tex]\(\theta\)[/tex]-axis (x-axis) from [tex]\(-π\)[/tex] to [tex]\(π\)[/tex].
- Draw the [tex]\(f(\theta)\)[/tex]-axis (y-axis) which typically extends up and down symmetrically.
- Plot the vertical asymptotes at [tex]\(\theta = -\frac{\pi}{2} \)[/tex] and [tex]\(\theta = \frac{\pi}{2} \)[/tex] as dashed or dotted vertical lines.
- Between these asymptotes, sketch the curve of [tex]\( \tan(\theta) \)[/tex] starting at [tex]\( \theta = -\frac{\pi}{2} \)[/tex] (approaching from just left of the asymptote where [tex]\( \tan(\theta) \rightarrow -\infty \)[/tex]), crossing the origin at [tex]\( \theta = 0 \)[/tex], and then approaching the asymptote at [tex]\( \theta = \frac{\pi}{2} \)[/tex] (where [tex]\( \tan(\theta) \rightarrow +\infty \)[/tex]).
- Extend the same pattern from [tex]\(-π\)[/tex] to [tex]\(-\frac{\pi}{2}\)[/tex] and from [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(π\)[/tex], mirroring the behavior around each interval.
6. Add Important Details:
- Label the [tex]\(\theta\)[/tex]-axis with the points [tex]\(-\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi\)[/tex].
- Plot horizontal gridlines for easy visualization of how [tex]\( \tan(\theta) \)[/tex] behaves approaching [tex]\( \pm\infty \)[/tex].
Here is a rough sketch of how the graph would look:
```
|
10 +
|
| + + +
| ⋰ /
| / ⋱
5 + // \\
| / \
| ⋱ ⋰
| / ⋱
| / \
0 +----|--------|--------|--------|--------|---- θ
-π -π/2 0 π/2 π
| \ / /
| ⋱ ⋲ ⋰
| \ / ⋲
| \ // ⋲
-5 + \ // ⋲
| \ /
| ⋱ ⋰
| \ /
-10 + ⋱ ⋰
| ⋲ /
|
```
### Conclusion
- The graph of [tex]\( f(\theta) = \tan(\theta) \)[/tex] on the interval [tex]\([-π, \pi]\)[/tex] features smooth, periodic behavior with vertical asymptotes at [tex]\(\theta = -\frac{\pi}{2} \)[/tex] and [tex]\(\theta = \frac{\pi}{2} \)[/tex].
- Understanding these key points and behavior of the tangent function is critical in graphing it correctly.
