Problem 42 - Precalculus with Limits

Question

In Exercises 37-54, a point in rectangular coordinates is given. Convert the point to polar coordinates. \(\left(3, 0\right)\)

Step-by-Step Solution

Answer

The polar coordinates for the point (3,0) are \(r = 3\) and \(θ = 0\).

Step 1: Identify the given coordinates

Identify the rectangular coordinates for the point, which in this case are \(x = 3\) and \(y = 0\).

Step 2: Apply the formula for r

Calculate the polar coordinate \(r\), which is given by the formula \(r = \sqrt{x^2 + y^2}\). Substitute \(x = 3\) and \(y = 0\) into the equation.

Step 3: Calculate r

Upon substitution, the formula becomes \(r = \sqrt{3^2 + 0^2}\), simplifying to \(r = 3\).

Step 4: Apply the formula for θ

Calculate the polar coordinate \(θ\), which is given by the formula \(θ = arctan (y/x)\). Here, \(θ = arctan (0/3)\).

Step 5: Express the final answer

The expression simplifies to \(θ = 0\). So the polar coordinates are \(r = 3\) and \(θ = 0\).

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative method to describe the location of a point in a two-dimensional plane. Unlike rectangular coordinates, which use horizontal and vertical distances (x and y coordinates), polar coordinates represent positions based on a radius and an angle relative to a reference direction (usually the positive x-axis).

In polar coordinates, a point is described by a pair \( (r, \theta) \) where \( r \) is the distance from the origin to the point, and \( \theta \) is the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point. Polar coordinates are particularly useful in scenarios involving circular or spiral shapes where distances from a central point are more relevant than horizontal or vertical positioning.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common nomenclature used to represent points in geometry. In this system, two perpendicular lines, usually labeled as the x-axis (horizontal) and y-axis (vertical), intersect at a point called the origin.

A point in this coordinate system is represented by the ordered pair \( (x, y) \), where \( x \) indicates the horizontal distance from the y-axis, and \( y \) indicates the vertical distance from the x-axis. The sign of the coordinates indicates the direction: positive \( x \) values are to the right of the origin, positive \( y \) values are above the origin, negative \( x \) values are to the left, and negative \( y \) values are below.

Coordinate Conversion

Converting between rectangular and polar coordinates involves using relationships derived from basic trigonometry and the Pythagorean theorem. The conversion process allows us to seamlessly switch between these two systems depending on which is more convenient for the problem we are working to solve.

To convert from rectangular to polar coordinates, we use the following formulas: \( r = \sqrt{x^2 + y^2} \) for the radius and \( \theta = arctan \left(\frac{y}{x}\right) \) for the angle, where \( x \) and \( y \) are the rectangular coordinates. It's essential to note that when finding \( \theta \), we often need to consider the quadrant of the original point to make an appropriate adjustment to the angle.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. Fundamental to coordinate conversion, trigonometry provides us the tools to translate circular motion and patterns into algebraic expressions and vice versa.

When converting coordinates, the trigonometric functions sine, cosine, and tangent are especially crucial. The tangent function, given by \( tan(\theta) = \frac{opposite}{adjacent} \), is used to find the angle \( \theta \) in polar coordinates when \( x \) and \( y \) are known. Understanding the unit circle and the special triangles (45°-45°-90° and 30°-60°-90°) aids in simplifying trigonometric calculations in coordinate conversions.