Problem 41 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ x=y $$

Step-by-Step Solution

Answer

The polar form of the equation is \( \theta = \frac{\pi}{4} + n\pi \).

Step 1: Understand Cartesian Coordinates Conversion

We're given the equation in Cartesian coordinates as \( x = y \). To convert this to polar coordinates, remember the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). We'll use these to rewrite the equation.

Step 2: Substitute Cartesian to Polar Relations

Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the given equation. This gives us the equation \( r \cos \theta = r \sin \theta \).

Step 3: Simplify the Equation

We have \( r \cos \theta = r \sin \theta \). Assuming \( r eq 0 \), we can divide both sides by \( r \) to simplify, resulting in \( \cos \theta = \sin \theta \).

Step 4: Solve for \( \theta \)

To solve \( \cos \theta = \sin \theta \), divide both sides by \( \cos \theta \) to get \( 1 = \tan \theta \). This means \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is an integer.

Step 5: Write the Polar Form

The equation in polar form is \( \theta = \frac{\pi}{4} + n\pi \). This represents a set of lines that pass through the origin at a 45° angle and every subsequent 180° increment.

Key Concepts

Cartesian coordinates

In mathematics, Cartesian coordinates are a way to specify the position of a point in a plane using a pair of numerical values, often represented as \( (x, y) \). These values denote the distance from the respective point to two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point determined by Cartesian coordinates corresponds to a unique position in a two-dimensional plane.

• The x-coordinate specifies how far along the point is from the y-axis.
• The y-coordinate indicates how far the point is from the x-axis.

Understandably, these coordinates help visualize and solve geometric and algebraic problems by breaking down the space into measurable units. In this framework, equations like \( x = y \) describe relations between these two dimensions.

conversion

The conversion from Cartesian coordinates to polar coordinates is essential when you need a different perspective or require a description based on angles and distance. In polar coordinates, a point in the plane is described using a radius \( r \) and an angle \( \theta \).
To convert any equation from Cartesian to polar form, use these essential equations:

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

This transformation leverages trigonometric relationships to transition from a grid-based system to a circular framework, making certain calculations more straightforward, especially when dealing with circular or periodic dynamics.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They play a crucial role in the process of converting and simplifying expressions during coordinate transformation.
In the exercise of converting \( x = y \) to polar form, the relation \( \cos \theta = \sin \theta \) emerges, leading us to use the identity related to tangent. Simplifying gives us:

• \( \tan \theta = 1 \) indicating that the angle \( \theta \) is \( \frac{\pi}{4} + n\pi \).

This is due to the basic identity that states \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Understanding these identities can be very useful in handling and converting equations from one form to another.

polar equations

Polar equations describe mathematical relationships using the polar coordinate system, where each point is defined by a distance from the origin and an angle from a reference direction. They often simplify the representation of curves and relations involving rotation and symmetry.
In polar form, an equation like \( \theta = \frac{\pi}{4} + n\pi \) represents a family of lines radiating out from the origin of the polar coordinate system.

• These lines have an equal spacing of \( 180^\circ \) between them.
• The periodicity in the angle \( n\pi \) allows the equation to express all lines that intersect with the origin at specified angles.

Transforming equations into polar form facilitates analysis and visualization, especially in circumstances where circular motion or radial symmetry is inherent.