Problem 24 - Precalculus with Limits

Question

In Exercises 19-28, a point in polar coordinates is given. Convert the point to rectangular coordinates. \(\left(1, 5\pi/4\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates equivalent to the polar coordinates (1, 5π/4) are \((- \sqrt{2}/2, - \sqrt{2}/2)\).

Step 1: Convert r and θ to x and y

The polar coordinates given are (1, 5π/4). Using the relationships \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\), we can substitute r = 1 and θ = 5π/4 to get the rectangular coordinates.

Step 2: Calculate x coordinate

First, we calculate the x coordinate: \(x = r \cdot \cos(\theta) = 1 \cdot \cos(5π/4) = - \sqrt{2}/2\).

Step 3: Calculate y coordinate

Next, we calculate the y coordinate: \(y = r \cdot \sin(\theta) = 1 \cdot \sin(5π/4) = - \sqrt{2}/2\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points on a plane using an

• ordered pair
• (x, y).

The x-coordinate specifies the horizontal position, while the y-coordinate specifies the vertical position. This system of coordinates is widespread in mathematics and everyday applications because it allows for straightforward representation of geometric figures.

When converting from polar to rectangular coordinates, it's helpful to remember that:

• the unit circle plays a significant role in trigonometrical values,
• and understanding the relationships between angles and coordinates is crucial for accurate calculations.

Rectangular coordinates are essential in graphing equations and performing calculations because they facilitate a clear geometric interpretation of algebraic expressions.

Coordinate Conversion

Coordinate conversion is the process of changing one set of coordinates into another.

• This is particularly useful in navigation, geometry, and physics,
• where different systems may simplify varying types of problems.

Converting polar coordinates to rectangular coordinates involves using foundational trigonometric relationships:

- The formula to find the x-coordinate is \[ x = r \cdot \cos(\theta) \]- The formula to find the y-coordinate is \[ y = r \cdot \sin(\theta) \]These formulas arise because any point's distance from the origin, represented by \( r \), forms a right triangle with the x and y axes. Here, \( \theta \) is the angle from the positive x-axis. By understanding these relationships, converting between coordinate systems becomes straightforward.

Trigonometric Functions

Trigonometric functions are essential in the world of coordinate conversion, especially when dealing with polar to rectangular transformations.

• The functions
• \( \cos(\theta) \) and \( \sin(\theta) \)

play an essential role in determining the respective x and y values from an angle \( \theta \).

These functions originate from the study of right triangles and the unit circle, which is

• a circle with radius 1 centered at the origin of a coordinate plane.

Here are the basic trigonometric identities helpful in coordinate conversion:- The cosine of an angle represents the horizontal distance on the unit circle, determining the x-coordinate.

\( \cos(\theta) = \text{adjacent}/\text{hypotenuse} \)

- The sine of an angle represents the vertical distance on the unit circle, determining the y-coordinate.

\( \sin(\theta) = \text{opposite}/\text{hypotenuse} \)

Understanding

• how to apply these functions
• and their relationships in coordinate conversion

is vital for anyone working with or learning about different coordinate systems.