Let's convert and write each of the given equations in rectangular form.
1. Conversion of [tex]\(\theta = -\frac{5 \pi}{6}\)[/tex] to rectangular coordinates:
In polar form, an angle [tex]\(\theta\)[/tex] and radius [tex]\(r\)[/tex] describe a point in the coordinate system. To convert this into rectangular form (i.e., [tex]\( (x, y) \)[/tex] coordinates), we use the following formulas:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
Assuming [tex]\(r = 1\)[/tex] (for the sake of this conversion):
- [tex]\(\theta = -\frac{5 \pi}{6}\)[/tex]
- [tex]\(\cos\left( -\frac{5 \pi}{6} \right) = -\frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\sin\left( -\frac{5 \pi}{6} \right) = -\frac{1}{2}\)[/tex]
Hence, the rectangular coordinates are:
[tex]\[
x = 1 \cdot \cos\left( -\frac{5 \pi}{6} \right) = -\frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
y = 1 \cdot \sin\left( -\frac{5 \pi}{6} \right) = -\frac{1}{2}
\][/tex]
Therefore, the rectangular form is:
[tex]\[
(-0.8660254037844387, -0.49999999999999994)
\][/tex]
2. Equation [tex]\(y = -\frac{\sqrt{3}}{3} x\)[/tex]:
This equation is already in rectangular form (linear form), where the slope [tex]\(m = -\frac{\sqrt{3}}{3}\)[/tex] and the intercept [tex]\(b = 0\)[/tex].
3. Equation [tex]\(y = \sqrt{3} x\)[/tex]:
This equation is also already in rectangular form, with a slope [tex]\(m = \sqrt{3}\)[/tex] and an intercept [tex]\(b = 0\)[/tex].
4. Equation [tex]\(y = \frac{\sqrt{3}}{3} x\)[/tex]:
This equation is already in rectangular form, with a slope [tex]\(m = \frac{\sqrt{3}}{3}\)[/tex] and an intercept [tex]\(b = 0\)[/tex].
5. Equation [tex]\(y = x\)[/tex]:
This equation is already in rectangular form, with a slope [tex]\(m = 1\)[/tex] and an intercept [tex]\(b = 0\)[/tex].
So, summarizing the solutions:
- [tex]\(\theta = -\frac{5 \pi}{6}\)[/tex] in rectangular form is:
[tex]\[
(-0.8660254037844387, -0.49999999999999994)
\][/tex]
- [tex]\(y = -\frac{\sqrt{3}}{3} x\)[/tex] remains as [tex]\(y = -\frac{\sqrt{3}}{3} x\)[/tex]
- [tex]\(y = \sqrt{3} x\)[/tex] remains as [tex]\(y = \sqrt{3} x\)[/tex]
- [tex]\(y = \frac{\sqrt{3}}{3} x\)[/tex] remains as [tex]\(y = \frac{\sqrt{3}}{3} x\)[/tex]
- [tex]\(y = x\)[/tex] remains as [tex]\(y = x\)[/tex]
These equations are thus expressed in their rectangular (Cartesian) forms.
