10. sketch the graph of the following polar equation.\n$$\\theta = -\\frac{\\pi}{3}$$
10. sketch the graph of the following polar equation.\n$$\\theta = -\\frac{\\pi}{3}$$
Answer
Explanation:
Step1: Identify the polar equation type
The equation $\theta = -\frac{\pi}{3}$ represents all points with a constant angle, regardless of the radius $r$.
Step2: Determine the geometric shape
In polar coordinates, an equation of the form $\theta = c$ represents a straight line passing through the pole (origin).
Step3: Convert to rectangular coordinates
Use the relationship $\tan(\theta) = \frac{y}{x}$ to find the slope $m$ of the line. $$m = \tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}$$
Step4: Describe the line's orientation
The line passes through the origin $(0,0)$ with a slope of $-\sqrt{3}$, extending through the second and fourth quadrants.
Step5: Sketching details
Draw a line through the origin making an angle of $60^{\circ}$ clockwise from the positive x-axis.
Answer:
The graph is a straight line passing through the origin (pole) with a slope of $-\sqrt{3}$. It makes an angle of $-\frac{\pi}{3}$ (or $300^{\circ}$) with the positive x-axis, extending infinitely through the second and fourth quadrants.