Problem 66 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$ r=6 \sec \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \(x=6\) and the graph of this equation is a vertical line passing through the point (6,0).

Step 1: Understand Polar to Rectangular Conversion

Understand that the polar coordinates \(r\) and \(\theta\) corresponds to the rectangular coordinates \((x,y)\) in the following way. The radial distance \(r\) is equal to the distance from the origin to the point \((x, y)\) and the angle \(\theta\) refers to the counterclockwise angle from the x-axis to the point \((x, y)\). Thus, we have \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).

Step 2: Convert the Equation

Let's start by replacing \(r\) with \(6 \sec \theta\) in the formulas for \(x\) and \(y\). Here, remember that \(\sec \theta = 1/ \cos \theta\). So we have \(x = r \cdot \cos(\theta) = 6 \sec \theta \cdot \cos \theta = 6\) and \(y = r \cdot \sin(\theta) = 6 \sec \theta \cdot \sin \theta = 6 \tan \theta\). Therefore the rectangular form of equation will be a vertical line passing through \((6,0)\)

Step 3: Graph the Rectangular Equation

Sketch the vertical line \(x=6\) on the graph. The line will pass through the x-axis at the point (6, 0). Since it's a vertical line, there's no variation in the x-coordinate, all points on the line will have an x-coordinate of 6.

Key Concepts

Polar Coordinates

Polar coordinates are a unique way to represent a point in the plane using a distance and an angle. Instead of the typical rectangular coordinates

• The distance is denoted by \(r\), which tells how far the point is from the origin.
• The angle \(\theta\) indicates the counterclockwise rotation from the positive x-axis to the point.

These coordinates are particularly useful in situations involving circular paths or when angular relationships are crucial. For instance, polar coordinates make it simpler to describe circular or spiral paths, as the radius \(r\) relates directly to distance from the center, and \(\theta\) provides the angle position.

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, express points in terms of \((x, y)\) positions on a grid.

• The \(x\)-coordinate indicates the horizontal position, moving right from the origin.
• The \(y\)-coordinate shows the vertical position, moving up from the origin.

These coordinates are more intuitive for plotting straight lines and standard shapes, like squares and rectangles. Converting from polar to rectangular coordinates involves using the relationships:

• \(x = r \cdot \cos(\theta)\)
• \(y = r \cdot \sin(\theta)\)

Understanding both coordinate systems enhances versatility in solving various mathematical problems.

Trigonometric Functions

Trigonometric functions are at the heart of converting between polar and rectangular coordinates. These functions relate angles to side lengths in right triangles.- **Sine and Cosine:** These are the main functions used in conversions.

• \(\cos(\theta)\) provides the adjacent side over hypotenuse.
• \(\sin(\theta)\) gives the opposite side over hypotenuse.

- **Secant and Tangent:** These are also integral in understanding the polar equation given.

• \(\sec(\theta)\) is the reciprocal of \(\cos(\theta)\), simplifying \(r = 6 \sec(\theta)\) to \(x = 6\).
• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

Using these relationships helps to translate polar equations into a rectangular form, making them ready for graphing on standard coordinate grids.