Problem 96 - Precalculus

Question

The use of a different system of coordinates simplifies many mathematical expressions and some calculations are performed in an easier way. The rectangular coordinates \((x, y)\) are transformed into polar coordinates by the equations $$x=r \cos \theta \quad \text { and } \quad y=r \sin \theta$$ where \(r=\sqrt{x^{2}+y^{2}}(r \neq 0)\) and \(\tan \theta=\frac{y}{x}(x \neq 0) .\) In polar coordinates, the equation of the unit circle \(x^{2}+y^{2}=1\) is just \(r=1\) In calculus, we use polar coordinates extensively. Transform the rectangular equation to polar form. $$x^{2}+y^{2}=4 x$$

Step-by-Step Solution

Answer

The polar form of the equation is \(r = 4 \cos \theta\).

Step 1: Identify the Rectangular Equation

The given rectangular equation is \(x^2 + y^2 = 4x\). This equation needs to be converted into polar coordinates.

Step 2: Use Polar Coordinate Definitions

Recall from the problem statement that in polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Thus, \(x^2 + y^2 = r^2\). Additionally, \(x = r \cos \theta\).

Step 3: Substitute Polar Definitions

Substitute \(x = r \cos \theta\) and \(x^2 + y^2 = r^2\) into the rectangular equation: \(r^2 = 4(r \cos \theta)\).

Step 4: Simplify the Polar Equation

Simplify the equation \(r^2 = 4r \cos \theta\) by dividing both sides by \(r\) (assuming \(r eq 0\)): \(r = 4 \cos \theta\).

Step 5: Conclusion

The rectangular equation \(x^2 + y^2 = 4x\) is transformed into the polar equation \(r = 4 \cos \theta\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are often referred to as Cartesian coordinates. They use two perpendicular axes, typically labeled as the x-axis and the y-axis. Each point is represented as

• \((x, y)\) in a plane which describes the distance from the origin along the x-axis and the y-axis.

These coordinates are incredibly useful for defining locations and plotting graphs on a two-dimensional plane. They form the basis for many mathematical concepts and are easy to visualize due to their straightforward system of horizontal and vertical measurements.

Coordinate Transformation

Coordinate transformation involves converting one set of coordinates to another. This is essential in converting between rectangular and polar coordinates.

• For rectangular to polar, use the transformations \(x = r \cos \theta\) and \(y = r \sin \theta\).
• To find \(r\), the radial distance, use \(r = \sqrt{x^2 + y^2}\).
• The angle \(\theta\) can be found using \(\tan \theta = \frac{y}{x}\).

This transformation allows for various calculations and expressions to become simpler, especially when dealing with curves and circular objects.

Trigonometric Functions

Understanding trigonometric functions is crucial in coordinate transformation, as they directly relate angles to side lengths in triangles.

• \(\cos \theta\) represents the adjacent side over the hypotenuse in a right triangle.
• \(\sin \theta\) represents the opposite side over the hypotenuse.
• \(\tan \theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\), or the opposite side over the adjacent side.

These functions are pivotal when using polar coordinates, as they relate the position of a point to its angle \(\theta\) from the positive x-axis.

Unit Circle

The unit circle is a fundamental tool in trigonometry and calculus. It is a circle with a radius of one unit, centered at the origin of a coordinate system.

• The equation for the unit circle in rectangular coordinates is \(x^2 + y^2 = 1\).
• In polar coordinates, the unit circle is simply \(r = 1\).

The unit circle helps in understanding the periodic nature of trigonometric functions and is used to derive the standard values of trigonometric functions at various angles. It is instrumental in analyzing cyclic phenomena.

Equation Conversion

Equation conversion refers to the process of transforming equations from one form into another, such as converting from rectangular to polar forms.

• Start with a rectangular equation, like \(x^2 + y^2 = k\) or other forms.
• Substitute \(x\) and \(y\) with their polar coordinate equivalents: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Simplify the resulting equation to express it in terms of \(r\) and \(\theta\).

For example, transforming \(x^2 + y^2 = 4x\) into polar form involves replacing \(x^2 + y^2\) by \(r^2\), leading to the polar equation \(r = 4 \cos \theta\). This conversion simplifies the representation and solves problems using polar principles.