Problem 97 - 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. $$ \left(4, \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (-2, 3.464).

Step 1: Identify the given coordinates

The polar coordinates given are (4, 2π/3). This means that R=4 and θ=2π/3.

Step 2: Convert the polar coordinates to rectangular coordinates

To convert the coordinates, use the formulas: x=R cos θ and y= R sin θ. Therefore, x=4 cos (2π/3), y=4 sin (2π/3).

Step 3: Calculate x and y using a graphing utility

Using a graphing utility or scientific calculator, find the values of x and y. Round the results to three decimal places.

Key Concepts

Rectangular Coordinates

Understanding rectangular coordinates is essential for converting from polar coordinates. In the rectangular coordinate system, each point in the plane is determined by an ordered pair of numbers, commonly known as (x, y). These numbers represent distances from two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

• The x-coordinate indicates how far left or right the point is from the y-axis.
• The y-coordinate shows how far up or down the point is from the x-axis.

Rectangular coordinates are very useful in many fields such as physics and engineering, as they make it easier to visualize and solve problems involving two-dimensional diagrams.

Trigonometric Conversion

Converting polar coordinates to rectangular coordinates involves using trigonometric functions like sine and cosine. Polar coordinates are given as (R, θ), where R is the radius or distance from the origin, and θ is the angle from the positive x-axis.
To convert:

• Use the formula for x: \( x = R \cos \theta \)
• Use the formula for y: \( y = R \sin \theta \)

For our problem, \( R = 4 \) and \( \theta = \frac{2\pi}{3} \). Finding these values helps in transitioning between different coordinate systems. This method of trigonometric conversion bridges both polar and rectangular systems, offering powerful tools for calculations.
The cosine and sine functions help to break down polar radius into horizontal and vertical components, respectively.

Graphing Utility

A graphing utility is a helpful tool for visualizing and calculating complex equations. It can be a digital application, like a graphing calculator or software, that assists in plotting points and curves based on mathematical expressions.
Using a graphing utility can simplify finding rectangular coordinates from polar coordinates as it:

• Quickly computes values for \( x \) and \( y \) using the trigonometric formulas.
• Automatically rounds results to required decimal places, e.g., three decimal places as in our example.

If you do not have access to a graphing utility, a scientific calculator that has trigonometric functions can still help you perform the conversions needed.

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. It's an alternative to degrees and is particularly convenient in mathematics as it leads to simpler and more natural formulas.
The full circle is \( 2\pi \) radians, equivalent to 360 degrees. This means that radians are based on the arc length of the unit circle.
The conversion from degrees to radians is given by the formula:

• Degrees to radians: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
• Radians to degrees: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \)

In the example, \( \frac{2\pi}{3} \) radians is used, which corresponds to an angle in the unit circle. Understanding these conversions is key when working with angles in different mathematical contexts.