To find all values of [tex]\(\theta\)[/tex] in the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex] where [tex]\(r \cdot \sin \theta = 0\)[/tex], let's analyze the given condition step-by-step.
1. Understanding the Relationship:
The equation given is [tex]\(r \cdot \sin \theta = 0\)[/tex]. For this product to be zero, either [tex]\(r = 0\)[/tex] or [tex]\(\sin \theta = 0\)[/tex]. Since we are interested in values of [tex]\(\theta\)[/tex], we focus on the condition [tex]\(\sin \theta = 0\)[/tex].
2. Identifying [tex]\(\theta\)[/tex] Values:
We need to find the values of [tex]\(\theta\)[/tex] in the interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex] that satisfy [tex]\(\sin \theta = 0\)[/tex].
3. Sine Function Characteristics:
The sine function is zero at certain specific angles. Within one full cycle [tex]\(0 \leq \theta \leq 2\pi\)[/tex], [tex]\(\sin \theta\)[/tex] equals zero at:
- [tex]\(\theta = 0\)[/tex]
- [tex]\(\theta = \pi\)[/tex]
- [tex]\(\theta = 2\pi\)[/tex]
4. Conclusion:
Therefore, the values of [tex]\(\theta\)[/tex] within the given interval [tex]\(0 \leq \theta \leq 2\pi\)[/tex] that satisfy [tex]\(r \cdot \sin \theta = 0\)[/tex] are:
- [tex]\[
\theta_1 = 0
\][/tex]
- [tex]\[
\theta_2 = \pi
\][/tex]
- [tex]\[
\theta_3 = 2\pi
\][/tex]
5. Numerical Representation:
These specific angles correspond to the following approximate numerical values:
- [tex]\[
\theta_1 = 0
\][/tex]
- [tex]\[
\theta_2 \approx 3.141592653589793
\][/tex]
- [tex]\[
\theta_3 \approx 6.283185307179586
\][/tex]
Hence, the values of [tex]\(\theta\)[/tex] that satisfy the given condition are [tex]\(0, \pi,\)[/tex] and [tex]\(2\pi\)[/tex].
