Problem 98 - 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. $$ (5.2,1.7) $$

Step-by-Step Solution

Answer

The rectangular coordinates of the point are (approximately to three decimal places) \(x = -1.736, y = 5.06\).

Step 1 Convert \(\theta\) from degrees to radiants

For calculations, the angle \(\theta\) must be in radians. Convert \(\theta\) from degrees to radians using the conversion factor \(\frac{\pi}{180}\).

Step 2 Calculation of x-coordinate

Use the polar coordinate to find the rectangular x-coordinate. In this case, apply the conversion formula \(x = r \cos(\theta)\) which results in \(x = 5.2 \times \cos(1.7)\). Calculate the result by using a calculator or computing software.

Step 3 Calculation of y-coordinate

Use the polar coordinate to find the rectangular y-coordinate. Apply the conversion formula \(y = r \sin(\theta)\). In this case, it results in \(y = 5.2 \times \sin(1.7)\). Calculate the result by using a calculator or computing software.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, help us determine the exact position of a point on a plane. These coordinates are expressed as a pair \(x, y\), representing the distance along the x-axis and y-axis, respectively.
Such a system is based on perpendicular intersection lines, typically visualized as a grid or two-dimensional plane.

• The x-coordinate shows how far to the right or left of the y-axis the point is located.
• The y-coordinate displays how far above or below the x-axis the point sits.

Understanding how these rectangular coordinates relate to polar coordinates is crucial, as it allows us to seamlessly switch between different mathematical representations. Converting polar coordinates, such as (r, θ), into rectangular form involves applying trigonometric functions to determine the specific location of the point on the plane.

Converting Degrees to Radians

Angles in trigonometry can be expressed in degrees or radians. However, many mathematical computations, especially involving calculus and trigonometric functions, require angles to be in radians. The process of converting from degrees to radians is straightforward.
To convert an angle from degrees to radians, multiply it by \(rac{\pi}{180}\).
This conversion is vital in solving problems in polar coordinates since radians are the standard unit of angular measure in mathematics. For example, if given an angle of 1.7 degrees, you would calculate the radian equivalent by performing a simple multiplication, giving a result that integrates seamlessly into formulas and computations.

Trigonometric Functions

Trigonometric functions are what allow us to link the circular nature of polar coordinates to the linear framework of rectangular coordinates.
The key functions used here are cosine (\(\cos\)) and sine (\(\sin\)).
In the context of converting polar to rectangular coordinates:

• The cosine function helps calculate the x-coordinate by representing the horizontal distance from the origin. It is calculated as \(x = r \cos(\theta)\).
• The sine function helps find the y-coordinate, representing the vertical distance, using the formula \(y = r \sin(\theta)\).

These equations take an angle in radians and a radius from polar coordinates to smoothly transition into a Cartesian format, allowing for straightforward graph plotting.

Graphing Utility

A graphing utility is a powerful tool that assists in visualizing mathematical concepts, like the conversion from polar to rectangular coordinates. These utilities can be software programs or physical calculators equipped with advanced graphing functionalities.
A typical graphing utility can:

• Automatically convert angles from degrees to radians, simplifying calculations.
• Plot polar and rectangular coordinates on a Cartesian plane for better visualization.
• Provide decimal approximations quickly, which are essential when dealing with coordinates derived from trigonometric functions.

Such tools are extremely helpful in checking work and understanding the spatial relationships between different forms of coordinate points, especially when verifying calculations in a mathematical context.