Problem 39 - 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: Identify Rectangular Coordinates

The given rectangular coordinates are \((-6, 0)\). Here, \(x = -6\) and \(y = 0\).

Step 2: Calculate the Radius

The radius \(r\) is calculated using the formula \(r = \sqrt{x^2 + y^2}\). Substitute \(x = -6\) and \(y = 0\): \[ r = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6 \] Thus, \(r = 6\).

Step 3: Calculate the Angle

The angle \(\theta\) is determined using the arctangent function \(\theta = \tan^{-1}(\frac{y}{x})\). Since \(x = -6\) and \(y = 0\), \[ \theta = \tan^{-1}(0) = 0 \] However, since the point is on the negative x-axis, the angle should be adjusted to point in the correct direction, \[ \theta = \pi \].

Step 4: Write Polar Coordinates

Combine the calculated radius and angle to express the polar coordinates. The polar coordinates are \((r, \theta) = (6, \pi)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point in a plane using an ordered pair \(x, y\). In the rectangular coordinate system, the horizontal axis is labeled as the x-axis and the vertical axis as the y-axis. Each coordinate determines a specific distance from the origin along the axes:

• The x-coordinate corresponds to the point's horizontal position.
• The y-coordinate corresponds to the point's vertical position.

Given the coordinates \(-6, 0\), this indicates that the point is located 6 units to the left of the origin on the x-axis, with no vertical displacement. This placement along the axis makes conversion to polar coordinates quite straightforward.

Radius Calculation

The radius, often represented as \(r\), in polar coordinates refers to the distance from the origin to the point on the plane. To find this distance, we use the Pythagorean theorem, which is summarized in the formula: \[ r = \sqrt{x^2 + y^2} \] For the point \(-6, 0\), this calculation involves:

• Squaring the x-coordinate: \((-6)^2 = 36\)
• Squaring the y-coordinate: \(0^2 = 0\)

Adding these gives \(\sqrt{36 + 0} = \sqrt{36} = 6\). Therefore, the radius is 6 units. This radius measures the direct line distance from the origin to the point, affirming the notion that polar coordinates essentially rotate around a central point, making distance straightforward to determine.

Angle Determination

In polar coordinates, the angle \(\theta\) describes the direction from the positive x-axis to the line connecting the origin and the point. It is typically measured in radians, where \(0 \leq \theta < 2\pi\). The angle can be determined using the formula: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \] For the coordinates \(-6, 0\), substituting the values gives:

• \(\frac{0}{-6} = 0\)
• \(\tan^{-1}(0) = 0\)

In standard position, where a point lies on the negative x-axis like \(-6, 0\), the angle is adjusted to \(\pi\) radians instead of zero. This adjustment correctly orients the point along the negative x-direction, demonstrating how angles in polar coordinates correspond to standard directional benchmarks as found in geometry.