To convert the polar equation [tex]\(\theta = \frac{\pi}{3}\)[/tex] into rectangular form and identify the type of curve represented, we will follow these steps:
### Step 1: Understanding the Polar Equation
The given polar equation is:
[tex]\[
\theta = \frac{\pi}{3}
\][/tex]
This means that the angle [tex]\(\theta\)[/tex] is fixed at [tex]\(\frac{\pi}{3}\)[/tex] radians.
### Step 2: Polar to Rectangular Coordinates
In polar coordinates, (r, [tex]\(\theta\)[/tex]) can be converted to rectangular coordinates (x, y) using the following relationships:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
### Step 3: Substituting the Fixed Angle
For [tex]\(\theta = \frac{\pi}{3}\)[/tex], we use the known trigonometric values:
[tex]\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\][/tex]
[tex]\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Thus, substituting these values into the equations for x and y, we get:
[tex]\[
x = r \cdot \frac{1}{2}
\][/tex]
[tex]\[
y = r \cdot \frac{\sqrt{3}}{2}
\][/tex]
### Step 4: Eliminating [tex]\(r\)[/tex]
To eliminate [tex]\(r\)[/tex], we can divide the equation for [tex]\(y\)[/tex] by the equation for [tex]\(x\)[/tex]:
[tex]\[
\frac{y}{x} = \frac{r \cdot \frac{\sqrt{3}}{2}}{r \cdot \frac{1}{2}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
This simplifies to:
[tex]\[
y = \sqrt{3} \cdot x
\][/tex]
### Step 5: Rectangular Form and Curve Type
Hence, the rectangular form of the given polar equation is:
[tex]\[
y = \sqrt{3} \cdot x
\][/tex]
This equation represents a straight line passing through the origin (0,0) with a slope of [tex]\(\sqrt{3}\)[/tex]. Thus, the type of curve is a straight line.
Therefore, the rectangular form of the polar equation [tex]\(\theta = \frac{\pi}{3}\)[/tex] is:
[tex]\[
y = \sqrt{3} \cdot x
\][/tex]
And the type of curve represented is a straight line.
