Problem 72 - College Trigonometry

Question

60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \)r\( in all but \)\\# 60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for \)\theta$. $$ x^{2}+y^{2}=x $$

Step-by-Step Solution

Answer

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

Step 1: Convert Rectangular to Polar Form

The original equation is given in rectangular coordinates: \(x^2 + y^2 = x\). To convert this to polar coordinates, we use the relations between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\), which are: \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the equation.

Step 2: Substitute Polar Relationships

Substituting, we have \((r \cos \theta)^2 + (r \sin \theta)^2 = r \cos \theta\). Simplify the left side: \(r^2 \cos^2 \theta + r^2 \sin^2 \theta = r \cos \theta\). Using the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\), the equation becomes \(r^2 = r \cos \theta\).

Step 3: Simplify and Solve for \(r\)

Divide both sides of the equation by \(r\), provided \(r eq 0\), to get \(r = \cos \theta\). Note that \(r = 0\) could also be a solution, but this represents the trivial solution at the origin, which doesn't impact the polar equation structure.

Step 4: Conclude with the Polar Equation

The equation in polar coordinates is \(r = \cos \theta\). This represents the set of all points where the radius equals the cosine of the angle with the positive x-axis.

Key Concepts

Rectangular Coordinates

When dealing with rectangular coordinates, you typically work within the familiar Cartesian coordinate system using an ordered pair \((x, y)\).

• The x-coordinate represents the horizontal distance from the origin.
• The y-coordinate represents the vertical distance from the origin.

For example, the coordinate \((3, 4)\) tells us that the point is 3 units to the right of the y-axis and 4 units above the x-axis. Rectangular coordinates are standard in many areas of math because they describe positions using perpendicular lines.In mathematical exercises, converting from rectangular to other coordinate systems, like polar, often provides insights. This conversion involves understanding the relationships between the two systems, which we’ll cover shortly.

Trigonometric Identities

Trigonometric identities are essential tools in converting equations between coordinate systems. When working with angles, trigonometric functions like sine and cosine are used to express these relationships.

• Commonly used identities include \(\cos^2 \theta + \sin^2 \theta = 1\).
• This identity helps simplify expressions and transforms rectangular equations into polar form.

For example, in the conversion of the equation \(x^2 + y^2 = x\) to its polar form, we substitute \(x = r \cos \theta\) and \(y = r \sin \theta\). Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), the equation simplifies to \(r^2 = r \cos \theta\). Thus, trigonometric identities play a crucial role in manipulations and simplifications during conversions.

Coordinate Conversion

Coordinate conversion is the process of changing coordinates from one system to another. This is often done to simplify problems or to switch to a more convenient coordinate system for a given context. When converting from rectangular to polar coordinates, we use specific relationships:

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

In our exercise, after substituting \(x = r \cos \theta\) and \(y = r \sin \theta\) in the rectangular equation \(x^2 + y^2 = x\), we simplify using trigonometric identities to find the polar form. This conversion shows how the expression \(r = \cos \theta\) represents points in a circle-like manner, where the radius equals the cosine of the angle from the positive x-axis.