Problem 54 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-3,-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

(0, 3)

Step 1: Understanding Polar to Rectangular Coordinate Conversion

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

Step 2: Substitute Values into Formulas

Given the polar coordinates are \((-3, -\frac{\pi}{2})\). Substitute \(r = -3\) and \(\theta = -\frac{\pi}{2}\) into the formulas. \[ x = -3 \cos\left(-\frac{\pi}{2}\right) \] and \[ y = -3 \sin\left(-\frac{\pi}{2}\right) \].

Step 3: Calculate the Values of Cosine and Sine

Calculate the cosine and sine of \(-\frac{\pi}{2}\): \[ \cos\left(-\frac{\pi}{2}\right) = 0 \] and \[ \sin\left(-\frac{\pi}{2}\right) = -1 \].

Step 4: Compute the Rectangular Coordinates

Using the computed values: \[ x = -3 \cdot 0 = 0 \] and \[ y = -3 \cdot (-1) = 3 \]. Hence, the rectangular coordinates are \((0, 3)\).

Key Concepts

Coordinate Conversion

Coordinate conversion is essential when you need to switch between different systems of locating points. For instance, converting between polar and rectangular coordinates. This process involves specific formulas that relate the two systems. In polar coordinates, a point is identified by its distance from the origin (r) and its angle (\theta) from the positive x-axis.
In rectangular coordinates, a point is identified by its x and y values forming a right-angled triangle with the origin.
The conversion formulas are:

• \( x = r \cos(\theta)\)
• \( y = r \sin(\theta)\)

These equations help convert polar coordinates (\r, \theta\right) into rectangular coordinates (\right, \right>). This understanding is crucial in many fields like engineering, physics, and computer graphics, where different scenarios may require different coordinate systems.

Polar Coordinates

Polar coordinates are a way of representing points in a plane using distances and angles. Unlike the Cartesian system, which uses x and y coordinates, polar coordinates use a radius and an angle.

• **Radius (r)**: This is the straight-line distance from the point to the origin (0, 0).
• **Angle (\theta)**: This is the angle measured from the positive x-axis to the line segment connecting the origin to the point.

One key feature of polar coordinates is that they can represent the same point in multiple ways, simply by adding multiples of \(2\pi\) (360 degrees) to the angle. In some cases, the radius can even be negative, which means you measure the distance in the opposite direction of the angle.
For example, in the exercise, we have polar coordinates \((-3, -\frac{\pi}{2})\). The negative radius indicates we measure 3 units in the opposite direction of \(-\frac{\pi}{2},\). Understanding how to interpret these coordinates helps you solve conversion problems accurately.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common way to specify locations in a plane.
They rely on orthogonal axes—horizontal (x-axis) and vertical (y-axis)—to measure distances. The coordinates of a point are given as an ordered pair (x, y), which describes its distance from the origin (0, 0) along each axis.
Your daily maps use Cartesian coordinates to show locations by latitude and longitude.

• **X-coordinate**: Measures the horizontal distance from the origin.
• **Y-coordinate**: Measures the vertical distance from the origin.

Converting from polar to rectangular involves computing these distances using trigonometric functions, as seen in the exercise. For example, we calculated \( x = -3 \cdot 0 = 0 \) and \( y = -3 \cdot (-1) = 3 \) to find the rectangular coordinates (0, 3) from the given polar coordinates \((-3, -\frac{\pi}{2})\). Understanding rectangular coordinates is fundamental to grasping more complex mathematical and real-world concepts.