Problem 16 - Algebra and Trigonometry

Question

Different Polar Coordinates for the Same Point: Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\). (3,1)

Step-by-Step Solution

Answer

The point \((3, 1)\) can also be represented as \((-3, 4.1416)\) and \((3, 7.2832)\).

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the distance from the pole (origin), and \(\theta\) is the angle from the positive x-axis. The given coordinates are \((3, 1)\). Here, \(r = 3\) and \(\theta = 1\) radian.

Step 2: Plot the Point

To plot the point \((3, 1)\), start at the origin, move 3 units outward in the direction that makes an angle of 1 radian with the positive x-axis. This locates the point in the polar coordinate system.

Step 3: Find the Equivalent Polar Coordinate with Negative r

For the polar coordinates with \(r < 0\), we can use the coordinate \((-3, 1 + \pi)\). This means you move 3 units in the opposite direction by adding \(\pi\) radians (equivalent to 180 degrees). Thus, the new coordinates become \((-3, 4.1416)\).

Step 4: Confirm the Equivalent Polar Coordinate with Positive r

For \(r > 0\), we can find equivalent angles by adding multiples of \(2\pi\) to \(\theta\). Another representation is \((3, 1 + 2\pi)\), which results in \((3, 7.2832)\). This means you complete a full circle (\(2\pi\) radians) and then advance in the same direction as initially.

Key Concepts

Radians

Radians are a way of measuring angles, commonly used in mathematics and physics. Unlike degrees, which divide a circle into 360 parts, a radian measures the angle created when an arc of a circle is equal in length to the radius of the circle. This is a more natural approach when dealing with circular and rotational problems.

• One complete circle equals to \(2\pi\) radians, which is roughly 6.2832.
• An angle of \(\pi\) radians corresponds to a half-circle or 180 degrees.
• To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

When dealing with polar coordinates, radians provide a precise method to determine direction relative to the positive x-axis. Understanding radians is essential in navigating the polar plane.

Negative r

In polar coordinates, the radius \(r\) determines how far a point is from the origin or pole. Usually, we think of \(r\) as being positive, which means the point is located in the direction of the angle \(\theta\).
However, a negative \(r\) means the point is in the opposite direction. This can be counterintuitive but is crucial for flexibility in polar coordinates.

• If \(r\) is negative, shift the angle \(\theta\) by \(\pi\) (180 degrees) to find the equivalent positive position.
• This approach ensures we still cover the same point, just observed from another direction.
• Example: The polar coordinate \((-3, 1 + \pi)\) represents a point that is the same as \((3, 1)\), but located oppositely on the polar plane.

Negative \(r\) coordinates allow us to explore more ways of representing points, crucial for comprehensive polar geometry.

Angle Conversion

In polar coordinates, angles are often added or adjusted to create equivalent representations of points. Knowing how to convert or modify these angles is vital.
Angle conversions often involve adding or subtracting multiples of \(2\pi\) radians, which represents a full circle rotation.

• Adding \(2\pi\) to an angle brings us back to the same point on the plane, essentially creating equivalent angles.
• For example, the point \((3, 1)\) can also be represented as \((3, 1 + 2\pi)\).
• This is useful in contexts where angles might need to be adjusted for different periods or rotations in a problem.

Converting angles helps in finding various representations of the same polar point, an important skill for identifying position in polar coordinates.

Plotting Points

When plotting points in polar coordinates, the process differs from the Cartesian system. Here, you focus on the distance from a central point and the direction in relation to the positive x-axis.

• Start from the origin. Move outward by the radius \(r\).
• Ensure the direction of movement aligns with the angle \(\theta\), given in radians.
• Plotting accurately helps in visualizing and understanding the location of points on the graph.

Using our current example, to plot \((3, 1)\), move 3 units from the origin at an angle of 1 radian. This approach allows for comprehensive navigation and plotting on the polar plane.