Question

$$\text {Sketch the following sets of points.}$$ $$\\{(r, \theta): \theta=2 \pi / 3\\}$$

Short answer

Question: Sketch the set of points with the polar coordinates {(r, θ) : θ = 2π/3}. Answer: To sketch the set of points, draw a ray starting from the origin at an angle of 120° counterclockwise from the positive x-axis. This ray represents all points where θ = 2π/3, as r can vary from 0 to infinity.

Step-by-step solution

Finding the angle

The angle θ is given as a constant value of 2π/3. This angle can be converted into degrees by multiplying it by 180/π: $$\theta = \frac{2\pi}{3} \cdot \frac{180}{\pi} = 120°$$ So, the line will be at an angle of 120° counterclockwise from the positive x-axis.

Visualizing the line

As there are no restrictions on the value of r, the line extends indefinitely. Now that we know the angle, draw a line at 120° from the positive x-axis as a line passing through the origin. This line will include all points of the set.

Sketching the set of points

To sketch the points, draw a ray starting from the origin at an angle of 120° from the positive x-axis. Label the lines and indicate that r can vary from 0 to infinity. Here, you have sketched the set of points with the polar coordinates {(r, θ) : θ = 2π/3}.

Key concepts

Angle Conversion

Working with polar coordinates often requires converting angles from radians to degrees or vice versa. This is because while mathematicians and many scientific applications commonly use radians, degrees are more intuitive for most people. The conversion process is straightforward.

• To convert an angle from radians to degrees, multiply by \( \frac{180}{\pi} \).
• Conversely, to convert degrees to radians, multiply by \( \frac{\pi}{180} \).

For example, converting the angle \( \theta = \frac{2\pi}{3} \) radians to degrees involves multiplying by \( \frac{180}{\pi} \), which simplifies to 120°. Understanding this conversion helps you visualize angles better when sketching graphs.

Sketching Graphs

When sketching graphs in polar coordinates, it is essential to understand how the angle and radius define the position of points. Each set of points {(r, θ)} corresponds to a unique location on the polar plane. The angle \( \theta \) indicates direction, while \( r \) denotes how far from the origin (the center) the point lies.
In the case where \( \theta = \frac{2\pi}{3} \), it represents a line, not just a collection of points, because there is no restriction on \( r \). The line extends from the origin outwards at the given angle. To sketch these points:

• Start at the origin (0).
• Draw a line in the direction of \( \theta \), which is 120° from the positive x-axis.
• Ensure the line extends indefinitely, indicating \( r \) can vary from 0 to infinity.

It's a good practice to label your axes and the angle clearly to aid understanding.

Trigonometry

Understanding polar coordinates often involves using the basics of trigonometry. In polar graphs, the angle \( \theta \) and radius \( r \) are related to the Cartesian coordinates \( (x, y) \) by the equations:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

This trigonometric relationship allows for converting polar coordinates into Cartesian coordinates. In the context of this exercise, knowing \( \theta \) helps determine the line's orientation in polar coordinates.
By understanding the trigonometric function associated with the angle, we can predict the line's behavior when sketching. For example, knowing that \( \theta = 120° \) means strong reliance on the sine function in determining \( y \)-coordinates, due to the nature of angles in the second quadrant. Using trigonometry, you not only sketch accurately but also translate between coordinate systems seamlessly.