Problem 74 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ y^{2}=2 x $$

Step-by-Step Solution

Answer

The polar form is \( r = \frac{2 \cos\theta}{\sin^{2}\theta} \).

Step 1 - Express x and y in terms of r and θ

Recall the polar coordinates conversions: \[ x = r \cos\theta \] and \[ y = r \sin\theta \]

Step 2 - Substitute x and y in the given equation

The given equation is \[ y^{2} = 2x \] Substitute \( x = r \cos\theta \) and \( y = r \sin\theta \) into this equation: \[ (r \sin\theta)^{2} = 2 (r \cos\theta) \]

Step 3 - Simplify the equation

First, simplify the left side: \[ r^{2} \sin^{2}\theta = 2r \cos\theta \] Then, divide both sides by \( r \) (assuming \( r eq 0 \)): \[ r \sin^{2}\theta = 2 \cos\theta \]

Step 4 - Solve for r

Finally, solve for \( r \): \[ r = \frac{2 \cos\theta}{\sin^{2}\theta} \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to locate points on a plane using two numbers: x and y. These coordinates are based on a simple grid system. The x-coordinate tells you how far to move left or right from the origin (the point (0,0)), and the y-coordinate tells you how far up or down to move.
For example, a point with coordinates (3, 4) is 3 units to the right and 4 units up from the origin. This coordinate system is very intuitive and is the foundation for many problems in mathematics and physics.

• x: Horizontal distance from the origin.
• y: Vertical distance from the origin.

Understanding rectangular coordinates is essential for moving on to more complex systems, such as polar coordinates.

Polar Coordinates

Polar coordinates represent points using a distance from the origin and an angle from the positive x-axis. This system is very useful for dealing with problems involving circles or angles.
A point in polar coordinates is expressed as \( (r, \theta) \), where:

• r: Distance from the origin (always non-negative).
• \theta: Angle in radians or degrees measured counterclockwise from the positive x-axis.

For instance, the point (5, \( \frac{\pi}{4} \)) in polar coordinates means moving 5 units away from the origin at an angle of 45 degrees. This system is particularly useful in physics and engineering for describing circular motions and waves.

Coordinate Conversion

To switch between rectangular and polar coordinates, you use conversion formulas. These formulas are based on trigonometric functions because they relate the distances and angles in both systems.
For converting from rectangular to polar coordinates:

• \( r = \sqrt{x^2 + y^2} \): Calculates the distance from the origin.
• \( \theta = \tan^{-1}(\frac{y}{x}) \): Determines the angle.

For converting from polar to rectangular coordinates:

• \( x = r \cos\theta \): Finds the x-coordinate.
• \( y = r \sin\theta \): Finds the y-coordinate.

These conversions are vital for solving many math problems, such as the given exercise. By knowing both systems, you can easily switch between them and find solutions to various types of problems.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides and are foundational in many areas of mathematics. The most commonly used functions in coordinate conversion and analysis are sine (sin), cosine (cos), and tangent (tan).
For a given angle \( \theta \):

• \( \sin\theta = \frac{opposite}{hypotenuse} \)
• \( \cos\theta = \frac{adjacent}{hypotenuse} \)
• \( \tan\theta = \frac{opposite}{adjacent} \)

In the exercise, to convert the given rectangular equation \( y^2 = 2x \) into polar form, we used:

• \( y = r \sin\theta \)
• \( x = r \cos\theta \)

Substituting these into the rectangular equation helped us find the polar form of the equation.
Understanding how to apply these trigonometric functions helps you master coordinate conversions and solve various math problems effectively.