Problem 11 - Thomas Calculus

Question

Find a polar equation in the form \(r \cos \left(\theta-\theta_{0}\right)=r_{0}\) for each of the lines in Exercises \(9-12\) . $$ y=-5 $$

Step-by-Step Solution

Answer

Polar equation: \( r \cos(\theta - \frac{\pi}{2}) = -5 \).

Step 1: Identify the Line Equation

The given line equation is in the Cartesian coordinate system: \( y = -5 \). This represents a horizontal line which is 5 units below the x-axis.

Step 2: Express in General Cartesian Form

Any equation of a line can be expressed in the form \( Ax + By = C \). For \( y = -5 \), this becomes \( 0 \cdot x + 1 \cdot y = -5 \). Hence, \( A = 0, B = 1, C = -5 \).

Step 3: Convert the Line Equation to Polar Form

In polar coordinates, the relation between Cartesian and polar is given by:\( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute these into the Cartesian equation: \( 0 \cdot (r \cos \theta) + 1 \cdot (r \sin \theta) = -5 \). This simplifies to \( r \sin \theta = -5 \).

Step 4: Express in Required Polar Equation Form

The required form is \( r \cos(\theta - \theta_0) = r_0 \). Use the identity: \( \cos(\theta - \theta_0) = \cos\theta \cos\theta_0 + \sin\theta \sin\theta_0 \). In \( r \sin \theta = -5 \), choose \( \theta_0 = \frac{\pi}{2} \) as this makes \( \sin\theta \) equivalent to \( \cos(\theta - \frac{\pi}{2}) \). Thus, the equation becomes \( r \cos(\theta - \frac{\pi}{2}) = -5 \).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a fundamental part of mathematics and are used to determine a point's position on a plane. The system uses two perpendicular axes, usually labeled as the x-axis and y-axis, to define the coordinates of any given point. Each point is represented by a pair of numerical values:

• x-coordinate: This is the horizontal value that shows how far away the point is from the y-axis. Positive values are to the right, and negative values are to the left.
• y-coordinate: This is the vertical value that shows how far away the point is from the x-axis. Positive values are above, and negative values are below the axis.

The equation "y = -5" is in Cartesian form, where y is -5, meaning all points on this line are situated horizontally 5 units below the x-axis. This straight, simple representation is one of the reasons Cartesian coordinates are popularly used in geometry and graphing.

Polar Equations

Polar equations offer a different and sometimes more intuitive way to represent points and curves, especially those involving angles or circular patterns. Unlike Cartesian coordinates that use a grid system, polar coordinates rely on a center point (the pole) and an angle from a reference direction:

• r: The radius or distance from the pole (origin) to the point.
• \( \theta \): The angle measured from a fixed direction, usually the positive x-axis in counterclockwise direction.

To convert from the Cartesian to polar system, equations often involve trigonometric functions. For instance, the given line "y = -5" in the polar form becomes \( r \sin \theta = -5 \). Here, \( r \) represents the distance from the origin, while \( \theta \) represents the angle.
The polar equation is eventually expressed in a special form: \( r \cos(\theta - \theta_0) = r_0 \). This form often uses trigonometric identities to switch between angle components, providing a smooth conversion and representation.

Trigonometric Identities

Trigonometric identities are essential in transforming and simplifying equations, particularly when working with polar coordinates. These identities are formulas involving trigonometric functions, like sine and cosine, that always hold true. Here are some key identities:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle addition formula: \( \begin{align*} \cos(a \pm b) &= \cos a \cdot \cos b \mp \sin a \cdot \sin b \ \sin(a \pm b) &= \sin a \cdot \cos b \pm \cos a \cdot \sin b \end{align*}\)

In our exercise, we use the identity \( \cos(\theta - \theta_0) = \cos \theta \cos \theta_0 + \sin \theta \sin \theta_0 \) to aid the conversion of the line equation into polar form. By choosing \( \theta_0 = \frac{\pi}{2} \), the equation \( r \sin \theta = -5 \) is transformed to \( r \cos(\theta - \frac{\pi}{2}) = -5 \), completing the process of expressing a Cartesian line in a polar equation format.