Problem 32 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (0,13 \pi) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (0, 0).

Step 1: Understanding Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured in radians from the positive x-axis.

Step 2: Identify the Radius and Angle

For the polar coordinates \((0, 13\pi)\), recognize that \(r = 0\) and \(\theta = 13\pi\).

Step 3: Apply the Conversion Formulas

The formulas to convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\) are: \[ x = r \cdot \cos(\theta) \] \[ y = r \cdot \sin(\theta) \]

Step 4: Calculate the Rectangular Coordinates

Substitute \(r = 0\) and \(\theta = 13\pi\) into the conversion formulas:- \(x = 0 \cdot \cos(13\pi) = 0\)- \(y = 0 \cdot \sin(13\pi) = 0\)

Step 5: Conclusion on Rectangular Coordinates

Regardless of the angle \(\theta\), since the radius \(r = 0\), the rectangular coordinates are always \((0,0)\) because the point is at the origin.

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe a point in a plane using a distance and an angle from a fixed point. Imagine standing at the center of a circle, the origin. Then, take a step outwards (that's the radius, or r) and pivot around (that's the angle, or \(\theta\)). Polar coordinates are expressed as \((r, \theta)\).

• \(r\) represents the distance from the origin to the point.
• \(\theta\) is the angle from the positive x-axis, usually measured in radians.

In our example, the polar coordinates \((0, 13\pi)\) mean our radius is 0. This means we don't actually step away from the origin, regardless of the angle. Consequently, the point is at the origin (0,0).

Rectangular Coordinates

When we talk about rectangular coordinates, we refer to the more familiar \((x, y)\) system you might already know. These coordinates plot a point's horizontal and vertical displacement from the origin on a coordinate plane.

• \(x\) measures how far the point is horizontally from the y-axis.
• \(y\) measures how far up or down the point is vertically from the x-axis.

In rectanglular coordinates, you move a certain distance along the x-axis, then "climb" to the point along the y-axis. For the polar coordinates \((r, \theta) = (0, 13\pi)\), the rectangular coordinates are \((x, y) = (0, 0)\) because the distance \(r\) is zero.

Coordinate Conversion

Conversion between polar and rectangular coordinates is an essential skill in mathematics, especially in trigonometry and navigation.

• To convert from polar to rectangular coordinates, use the formulas: \[ x = r \cdot \cos(\theta) \] \[ y = r \cdot \sin(\theta) \]
• For our exercise \((r,\theta) = (0, 13\pi)\), the values compute to \(x = 0 \cdot \cos(13\pi) = 0\) and \(y = 0 \cdot \sin(13\pi) = 0\).

Because \(r = 0\), any angle \(\theta\) still results in the point being at the origin (0,0), showcasing that in this case, the conversion leads to simplicity.

Trigonometry

Trigonometry is a field of mathematics that studies the relationships between side lengths and angles of triangles. It's the essential backbone needed for converting polar to rectangular coordinates.

• The cosine and sine functions often used in conversions are trigonometric functions derived from a right triangle.
• They relate an angle to the ratio of two sides in a right triangle.

When converting polar coordinates like \((0, 13\pi)\), we use trigonometric identities to transpose \(\theta\) into understandable x and y movements. However, with \(r = 0\), the results \(x = 0\) and \(y = 0\) arise from the multiplication by zero, demonstrating the power of trigonometry in a simple but authoritative way.