Problem 41 - Algebra and Trigonometry

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (-6,0) $$

Step-by-Step Solution

Answer

The polar coordinates are \((6, \pi)\).

Step 1: Find the Radius

The radius in polar coordinates, denoted as \( r \), is the distance from the origin to the point \((-6, 0)\). This is simply the absolute value of the \( x \)-coordinate since \( y=0 \). Thus, \( r = |-6| = 6 \).

Step 2: Determine the Angle

The angle \( \theta \) in polar coordinates is the angle formed with the positive \( x \)-axis. For the point \((-6, 0)\), which lies on the negative \( x \)-axis, \( \theta \) is \( \pi \) radians, as \( x \) is negative, and \( y \) is zero.

Step 3: Write the Polar Coordinates

With \( r = 6 \) and \( \theta = \pi \), the polar coordinates of the point \((-6, 0)\) are \( (6, \pi) \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to specify a location on a plane using an ordered pair of numbers. These numbers correspond to distances along perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical). For example, the point (-6, 0) indicates that we move 6 units to the left of the origin (0, 0) on the x-axis and not move on the y-axis at all.

• The first value in the pair, -6 in this case, tells us how far to move along the x-axis.
• The second value, 0, tells us how far to move up or down the y-axis.

By using these two values together, we can accurately plot any point on the plane.

Angle Conversion

Converting from rectangular to polar coordinates involves determining an angle, θ, which is the angle formed with the positive x-axis.

• If a point falls on the negative x-axis, like (-6, 0), the angle must be carefully calculated.
• Angles on the negative x-axis are directly opposite those on the positive side, meaning they measure π radians.

This ensures that the angle accurately represents the direction of the point from the origin.

Radius Calculation

The radius, represented by r, is a crucial part of polar coordinates. It shows the distance from the origin to our point on the plane. For points on the x-axis, the radius is just the absolute value of the x-coordinate.

• In our example, the point (-6, 0), we find the radius by calculating | -6 | = 6.
• This distance remains positive as it represents the magnitude of the position from the center.

Thus, the radius here tells us that despite the negative direction, the point is 6 units away from the origin.

Radians

Radians are a way of measuring angles, where the angle is described in terms of the radius of a circle. In mathematics, radians are often used over degrees because they allow for more straightforward integration into mathematical formulas and physics.

• One complete revolution around a circle is 2π radians.
• A straight angle, or half a circle, is equated to π radians, as seen in our calculation for the point on the negative x-axis.

By expressing the angle as π radians, we adhere to the standard conventions used in many mathematical calculations beyond just polar coordinates.