Problem 46 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ (-3, \pi) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(3, 0\).

Step 1 - Understanding Polar Coordinates

In polar coordinates, a point is represented as \(r, \theta\), where \(r\) is the radius or distance from the origin and \(\theta\) is the angle measured from the positive x-axis. For the point given as \((-3, \pi)\), \(r = -3\) and \(\theta = \pi\).

Step 2 - Converting to Rectangular Coordinates

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]

Step 3 - Calculate the x Coordinate

Substitute \(r = -3\) and \(\theta = \pi\) into the formula for \(x\): \[ x = -3 \cos(\pi) \] \[-3 \cos(\pi) = -3(-1) = 3\]

Step 4 - Calculate the y Coordinate

Substitute \(r = -3\) and \(\theta = \pi\) into the formula for \(y\): \[ y = -3 \sin(\pi) \] \[ -3 \sin(\pi) = -3(0) = 0 \]

Step 5 - Write the Rectangular Coordinates

The rectangular coordinates \((x, y)\) are obtained as \(3, 0\).

Key Concepts

Polar Coordinates

Polar coordinates provide a way to describe the position of a point in a plane using a radius and an angle. The radius, denoted as \(r\), is the distance from the origin to the point. The angle, denoted as \(\theta\), is measured from the positive x-axis to the point. These measures together help locate a point in a 2D plane. Polar coordinates are written as \((r, \theta)\). This system is particularly useful in scenarios involving rotations or circular motion.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point's position using two values: \(x\) and \(y\). The \(x\) value indicates how far the point is along the horizontal axis, and the \(y\) value indicates the distance along the vertical axis. These coordinates are written as \((x, y)\) and are very intuitive for most people because they directly map onto physical distances along perpendicular axes, making them very useful for plotting and visualizing points on a graph.

Coordinate Conversion

Converting from polar to rectangular coordinates involves assigning a position in rectangular coordinates that corresponds to the same point in polar coordinates. To make this conversion, you can utilize the trigonometric functions cosine and sine. The formulas used are:
\[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]
By substituting the given values of \(r\) and \(\theta\) into these equations, you can find the equivalent \(x\) and \(y\) coordinates.

Trigonometric Functions

Trigonometric functions like cosine (\(\cos\)) and sine (\(\sin\)) are essential for converting between polar and rectangular coordinates.

• \(\cos(\theta)\) gives the horizontal coordinate when multiplied by the radius \(r\).
• \(\sin(\theta)\) provides the vertical coordinate when multiplied by the radius \(r\).

For example, if \(\theta = \pi\) and \(r = -3\) as in the given exercise, we use:
\[ x = -3 \cos(\pi) = -3(-1) = 3 \] \[ y = -3 \sin(\pi) = -3(0) = 0 \]
This process helps us determine that the rectangular coordinates are \((3, 0)\). Trigonometric functions thus serve as a crucial bridge between the polar and rectangular coordinate systems.