Problem 31 - Precalculus

Question

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(2,2.74)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the point are (-1.664, 1.394)

Step 1: Identify the given polar coordinates

We can see the given polar coordinates are (2, 2.74), where 2 is the radius (r) and 2.74 is the angle in radians (θ)

Step 2: Apply the polar to rectangular conversion formulas

We can calculate the rectangular coordinates x and y by plugging r and θ into the equations x = rcos(θ) and y = rsin(θ). So, to get the x-coordinate, we multiply the radius by the cosine of the angle, and to get the y-coordinate, we multiply the radius by the sine of the angle.

Step 3: Compute the rectangular coordinates

By substituting r=2 and θ=2.74 into the equations, we get \(x = 2 \cdot \cos(2.74) = -1.664\) and \(y = 2 \cdot \sin(2.74) = 1.394\). These values are the rectangular coordinates of the given point. Note that the results were rounded to three decimal places.

Key Concepts

Polar Coordinates

Polar coordinates are a way to locate a point in a plane using two values: the radius and the angle. Imagine a circle centered at the origin of a graph. The point you want to locate will be somewhere along the radius of this circle.
The distance from the origin to the point is called the radius, or \( r \).

• The angle formed with the positive \( x \)-axis, measured in radians (or sometimes degrees), is referred to as \( \theta \).
• The format for polar coordinates is \((r, \theta)\), where \( r \) is the length of the line from the origin to the point, and \( \theta \) is the angle.

These coordinates are particularly useful in scenarios where a circular motion or direction-related measurement is involved. They provide an alternative method to the traditional way of marking points purely using horizontal and vertical lines, which is known as rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates refer to the standard way of plotting points using two dimensions commonly known as \( x \) and \( y \). Think of them like street grids in a city, where each point can be reached using a straight horizontal and a vertical line.
The axes meet at the origin, \((0,0)\). The point is thus defined by how far along the \( x \)-axis and \( y \)-axis it is found.

• The \( x \)-coordinate tells us how far right or left to go from the origin, making use of positive or negative values respectively.
• Similarly, the \( y \)-coordinate describes how far up or down to go, also using positive for up and negative for down.

In many scenarios, especially those not involving circular paths, rectangular coordinates offer a straightforward and intuitive way to describe locations. They are the primary mode of description in Cartesian coordinate systems and form the foundation for much of geometry and algebra.

Trigonomic Functions

Trigonomic functions play a crucial role in converting between polar and rectangular coordinates. These functions, primarily sine and cosine, help relate angles in a circle to lengths.

• The cosine function, \( \cos(\theta) \), relates to the \( x \)-coordinate, projecting the length along the horizontal axis.
• The sine function, \( \sin(\theta) \), correlates to the \( y \)-coordinate, determining the length along the vertical axis.

In the context of our problem, these functions transform the circular measures of polar coordinates into the linear measure of rectangular coordinates. By knowing the angle and the radius, we can apply:

• \( x = r \cdot \cos(\theta) \) which computes how far the point stretches horizontally.
• \( y = r \cdot \sin(\theta) \) which calculates its vertical stretch.

The use of radians is standard in these calculations because they are a direct measure related to the length of arcs in the unit circle, making the math cleaner and reducing potential errors through conversions between radians and degrees.