Problem 26 - Precalculus

Question

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(-2,7 \pi / 4)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the point \((-2, 7 \pi / 4)\) in polar coordinates are \((- \sqrt{2}, \sqrt{2})\).

Step 1: Identify the coordinates

Identify the polar coordinate that needs to be converted: \((-2, 7 \pi / 4)\). In this polar coordinate, \(r = -2\) and \(\theta = 7 \pi / 4\).

Step 2: Calculate x-coordinate

Calculate the x-coordinate using the formula \(x = r \cdot \cos(\theta)\). So, \(x = -2 \cdot \cos(7\pi / 4) = -2 \cdot \sqrt{2}/2 = -\sqrt{2}\).

Step 3: Calculate y-coordinate

Calculate the y-coordinate using the formula \(y = r \cdot \sin(\theta)\). So, \(y = -2 \cdot \sin(7\pi / 4) = -2 \cdot -\sqrt{2}/2 = \sqrt{2}\).

Key Concepts

Polar Coordinates

Understanding polar coordinates is essential for various fields, including mathematics, physics, and engineering. Unlike the familiar Cartesian or rectangular coordinates, which use horizontal and vertical distances to define a point's location, polar coordinates define a point based on its angle and distance from a central point, known as the origin.

In polar coordinates, a point is represented by a pair \( (r, \theta) \), where \( r \) is the radius or the distance from the origin, and \( \theta \) is the angle measured from the positive x-axis, usually in radians. The concept of negative radius, such as in the given exercise with \( r = -2 \), can be tricky. It implies the point is on the opposite side of the origin from where it would be with a positive radius along the same angle.

Rectangular Coordinates

Moving to rectangular coordinates, which you might know as x-y coordinates, makes use of a grid pattern to establish a point's position with two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

To define a point in this system, you give two values: \( x \) and \( y \) coordinates. These values represent the point's horizontal and vertical distances from the origin, respectively. Rectangular coordinates are widely used because they're intuitive for problems involving straight lines and right angles, often making them the first coordinate system students learn.

Coordinate Conversion

Coordinate conversion is a critical skill when dealing with problems that can be simplified or are only solvable in a different coordinate system. To convert from polar to rectangular coordinates, the formulas \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \) are used.

These formulas arise from the definition of sine and cosine in right-angled triangles, where \( r \) acts as the hypotenuse. It's important to pay attention to the given angle and whether it's in radians or degrees, as this will affect the calculation. The conversion allows one to leverage the strengths of both systems and offers more tools to approach a variety of mathematical tasks.

Trigonometry in Polar Coordinates

The trigonometry involved in the polar coordinate system often uses the sine and cosine functions to relate the polar and rectangular forms. In problems like our exercise, we can visualize the trigonometric circle, where the angle \( \theta \) defines a part of the circle, and the radius \( r \) determines how far along that angle the point lies.

When we calculated \( x = -2 \cdot \cos(7\pi / 4) \) and \( y = -2 \cdot \sin(7\pi / 4) \) in our exercise, we used the negative radius to reflect across the origin, tapping into the periodic nature of sine and cosine. This use of trigonometry is essential not only in coordinate conversion but also in understanding the relationship between the polar and rectangular systems.