Problem 99 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Use a graphing utility to find the rectangular coordinates of each point to three decimal places. $$ (-4,1.088) $$

Step-by-Step Solution

Answer

The rectangular coordinates of the point with polar coordinates (-4, 1.088) are approximately (-1.316, -3.834).

Step 1: Identify the polar coordinates

Identify the given polar coordinates. Here, the polar coordinates are (-4,1.088), where -4 is the radius (r) and 1.088 is the angle in radians (θ).

Step 2: Convert to rectangular coordinates

Use the formulas x = r*cos(θ) and y = r*sin(θ) to convert to rectangular coordinates. Calculate x = -4 * cos(1.088) and y = -4 * sin(1.088).

Step 3: Perform the calculations

Performing the calculations gives approximate x and y values to three decimal places.

Key Concepts

Polar Coordinates

Polar coordinates represent the location of a point in a plane using a radius and an angle relative to a fixed direction, typically the positive x-axis. In the format \( (r, \theta) \), \( r \) is the distance from the origin to the point, and \( \theta \) is the angle measured in radians from the positive x-axis to the line segment connecting the origin with the point.

The polar coordinate system is especially useful in scenarios where the relationship between two points is more easily expressed through angles and distances. For example, the orbits of celestial bodies are often described in polar coordinates. When given polar coordinates, like \( (-4, 1.088) \), the negative sign associated with the radius indicates that the point lies on the direct opposite side of the origin, along the line through the pole at an angle of \( 1.088 \) radians.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, identify the position of a point in a 2D space with respect to two perpendicular axes, generally labeled x (the horizontal axis) and y (the vertical axis). A point's location is determined by two numbers, \( (x, y) \), representing the horizontal and vertical distances from the origin.

The Cartesian coordinate system is the foundation for much of algebra and analytical geometry. It is particularly helpful when dealing with linear equations, parabolas, and other geometric shapes where uniform scales and right angles simplify the mathematics and visual representation.

Trigonometry in Polar to Rectangular Coordinates Conversion

The conversion from polar to rectangular coordinates requires the understanding of trigonometry. Specifically, the relationships of the sine and cosine functions to the unit circle. To convert a polar coordinate to rectangular form, you use the equations \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \), where \( r \) is the radius and \( \theta \) is the angle in radians.

Calculating x and y

Using the given polar coordinates \( (-4, 1.088) \) as an example, you'd calculate \( x \) as follows: \( x = -4 \cdot \cos(1.088) \) and \( y \) as \( y = -4 \cdot \sin(1.088)\). The negative radius essentially reflects the point across the origin, meaning you evaluate the cosine and sine at \( 1.088 \), and then reverse the sign of the resulting rectangular coordinates.

Graphing Calculator Usage

A graphing calculator is an invaluable tool in understanding and visualizing mathematical concepts from basic algebra to advanced calculus and statistics. For converting polar to rectangular coordinates, graphing calculators typically offer functions to directly perform this task. You can input the polar coordinates, and the calculator will apply the conversion formulas and display the rectangular coordinates.

Additionally, if you have to do the conversion manually, you can still use the calculator to compute the trigonometric values. For example, to find the rectangular coordinates of \( (-4, 1.088) \), you would enter the polar radius and angle into the calculator to find \( \cos(1.088) \) and \( \sin(1.088) \) to the required degree of precision, assisting you in obtaining an accurate conversion to rectangular coordinates.