Problem 67 - Thomas Calculus

Question

In Exercises \(65-68\) , find a polar equation for the given curve. In each case, sketch a typical curve. $$ x \cos \alpha+y \sin \alpha=p \quad(\alpha, p \text { constant }) $$

Step-by-Step Solution

Answer

The polar equation is \(r \cos(\theta - \alpha) = p\).

Step 1: Understand the given equation

The given equation is expressed in Cartesian coordinates: \(x \cos \alpha + y \sin \alpha = p\). In this equation, \(\alpha\) and \(p\) are constants, representing the angle and the perpendicular distance from the origin to the line, respectively.

Step 2: Convert Cartesian to Polar coordinates

Recall that in polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the given equation to convert it to polar form:\[r \cos \theta \cos \alpha + r \sin \theta \sin \alpha = p\]

Step 3: Factor out the common factor

Notice that \(r\) is a common factor in the two terms on the left-hand side. Factoring it out, we get:\[r (\cos \theta \cos \alpha + \sin \theta \sin \alpha) = p\]

Step 4: Apply the trigonometric identity

Use the trigonometric identity for the cosine of the sum of two angles: \(\cos(A + B) = \cos A \cos B + \sin A \sin B\). Applying this identity, the equation becomes:\[r \cos(\theta - \alpha) = p\]

Step 5: Express the polar equation

The equation \(r \cos(\theta - \alpha) = p\) is now expressed in polar coordinates, which describes a line at an angle \(\alpha\) from the x-axis and at a distance \(p\) from the origin.

Key Concepts

Cartesian Coordinates

In mathematics, Cartesian coordinates are a fundamental way to describe a point in space by specifying two or more values. These values determine the point's unique position on a plane or in space.

• In two-dimensional space, a point's position is depicted by \(x, y\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.
• In three-dimensional space, an additional \(z\) coordinate is used, allowing representation of a point in a three-dimensional grid.

This system allows for straightforward analysis of relationships between points and curves on a plane.

For our given equation, \(x \cos \alpha + y \sin \alpha = p\), we are specifying an equation of a line using Cartesian coordinates. Here, the constants \(\alpha\) and \(p\) are crucial in defining the specific orientation and position of the line relative to the origin. The understanding of Cartesian coordinates forms the baseline for converting this equation into another common system, namely polar coordinates.

Trigonometric Identities

Trigonometric identities are consistent mathematical statements involving trigonometric functions that hold true across various scenarios. They are essential tools in calculus and geometry for simplifying or transforming equations.

One of the identities used in this exercise is the cosine of a sum of angles: \(\cos(A + B) = \cos A \cos B + \sin A \sin B\). This identity helps rewrite equations in a more manageable form.

• These identities simplify the process of converting Cartesian equations to polar form or vice versa.
• They reveal equivalent expressions, allowing for alternative computations or interpretations of the same mathematical relationship.

In the solution, we see this identity come into play when expressing the Cartesian equation \(x \cos \alpha + y \sin \alpha = p\) in polar form. By applying the identity, our equation simplifies to \(r \cos(\theta - \alpha) = p\), where \(r\) and \(\theta\) are from the polar coordinate system. Trigonometric identities thus serve as bridges between different mathematical forms.

Equation Conversion

Equation conversion is a valuable skill that allows one to express mathematical relationships in different formats to ease analysis or computation.

Converting between Cartesian and polar coordinates involves using the relationships:

• In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\).
• This conversion provides an alternative viewpoint, particularly useful in problems involving radial symmetry or angles.

For the provided exercise, the goal was to convert the Cartesian equation into a polar form. By substituting polar coordinate expressions for \(x\) and \(y\), we reformulated the equation using polar terms, showing the effect of angle \(\alpha\) and distance \(p\) in a different light.

Equation conversion brings new insights and can simplify solving and visualizing mathematical problems by choosing a representation that best suits the context.