Problem 81 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The equation in polar coordinates is already \(r^2 = \cos \theta\).

Step 1: Recall Polar Coordinates Formulas

The polar coordinates \((r, \theta)\) are related to rectangular coordinates \(x\) and \(y\) by the formulas: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).

Step 2: Rewrite the Given Equation

The given equation is \(r^2 = \cos \theta\). This equation is already primarily in terms of polar coordinates.

Step 3: Verify If the Equation Follows Polar Form

The equation \(r^2 = \cos \theta\) matches the structure of polar form equations, thus there is no need for further conversion.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of representing a point in a plane using two values, typically labeled as \(x\) and \(y\). These coordinates are based on a Cartesian coordinate system, where \(x\) represents the horizontal distance from the origin, and \(y\) represents the vertical distance from the origin.

Rectangular coordinates are straightforward for plotting and visualizing points. For example, the point (3, 4) is located 3 units to the right of the origin and 4 units up. This system is particularly useful in many aspects of mathematics and engineering.

Points in rectangular coordinates can be converted to polar coordinates, and vice versa, using specific formulas which we'll discuss shortly.

Conversion Formulas

Conversion between rectangular coordinates \(x\) and \(y\) and polar coordinates \(r\) and \(\theta\) involves several key formulas.

To convert from rectangular to polar coordinates:

• \r = \sqrt{x^2 + y^2}\
• \(\theta\) = \tan^{-1}(\frac{y}{x})\

The first formula calculates the radius \(r\), which is the straight-line distance from the origin to the point. The second formula calculates the angle \(\theta\), which is measured counterclockwise from the positive x-axis to the line connecting the origin to the point.

Conversely, to convert from polar to rectangular coordinates:

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

These formulas help transition between coordinate systems, which is useful in various branches of science and engineering.

Trigonometric Relationships

Trigonometric relationships are fundamental in converting coordinates and understanding the geometry of different points.

The polar coordinates rely heavily on trigonometric functions: \( \sin \theta\) and \( \cos \theta\). These functions relate the angles and lengths of triangles to the coordinates of points.

For example, in the unit circle (a circle with radius 1 centered at the origin):

• \r=1\
• x = \cos \theta\
• y = \sin \theta\

Understanding these relationships helps in solving equations and converting between coordinate systems. They also form the basis for many mathematical and engineering applications, such as signal processing, wave analysis, and more.