Problem 22 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r=2 \sin \theta+2 \cos \theta$$

Step-by-Step Solution

Answer

The equation describes a circle with center (1, 1) and radius \(\sqrt{2}\).

Step 1: Convert Polar to Cartesian

The given polar equation is \( r = 2\sin\theta + 2\cos\theta \). We can convert it to a more familiar Cartesian form. Recall that \( x = r\cos\theta \) and \( y = r\sin\theta \). Also, remember that \( r^2 = x^2 + y^2 \) and \( r = \sqrt{x^2 + y^2} \). Re-express the given equation in terms of \( x \) and \( y \).

Step 2: Expand Both Sides

Multiply both sides by \( r \) to eliminate the fraction: \( r^2 = 2r\sin\theta + 2r\cos\theta \). Use \( r^2 = x^2 + y^2 \), \( y = r\sin\theta \), and \( x = r\cos\theta \) to get \( x^2 + y^2 = 2y + 2x \).

Step 3: Rearrange the Equation

Rearrange the terms: \( x^2 - 2x + y^2 - 2y = 0 \). This resembles a circle equation, and to identify its center and radius we need to complete the square.

Step 4: Complete the Square

Complete the square for both \( x \) and \( y \):For \( x \): \( x^2 - 2x \) becomes \( (x-1)^2 - 1 \).For \( y \): \( y^2 - 2y \) becomes \( (y-1)^2 - 1 \).Thus, the equation becomes: \((x-1)^2 + (y-1)^2 = 2\).

Step 5: Identify Circle's Characteristics

The circle equation \((x-1)^2 + (y-1)^2 = 2\) represents a circle centered at \((1, 1)\) with a radius of \(\sqrt{2}\).

Step 6: Sketch the Graph

To sketch the circle, mark the center at \((1, 1)\) on the Cartesian plane. Then, draw a circle with radius \(\sqrt{2}\), which is approximately 1.41 units.

Key Concepts

Cartesian Coordinates

In mathematics, Cartesian coordinates provide a way to define the position of a point in a two-dimensional plane using two numerical values. These values, typically denoted as \(x\) and \(y\), represent the point's horizontal and vertical positions, respectively. The coordinate system is named after René Descartes, who developed this form of graphing.
When working with polar equations, such as \(r = 2 \sin \theta + 2 \cos \theta\), we often convert them into Cartesian equations to take advantage of the familiarity and simplicity of the Cartesian plane.
To express a polar equation in Cartesian coordinates, apply the conversions: \(x = r \cos \theta\) and \(y = r \sin \theta\). Given that \(r = \sqrt{x^2 + y^2}\), these relationships allow us to reframe the problem in terms of \(x\) and \(y\), making it easier to analyze and graph.

Circle Equation

A circle in the Cartesian coordinate system is described by the equation \( (x-h)^2 + (y-k)^2 = r^2\), where \( (h, k)\) are the coordinates of the circle's center, and \( r \) represents its radius. This formula arises from the geometric definition of a circle as the set of all points equidistant from a given point, the center.
In the step-by-step solution, we transformed the polar equation into a Cartesian form that resembles a circle equation: \( (x-1)^2 + (y-1)^2 = 2 \).

• The center of this circle is at \( (1, 1)\), derived from the completed squares sections for both \( x \) and \( y \).
• The radius is \( \sqrt{2} \,\) computed from the right side of the equation.

Completing the square is essential in converting a general conic section equation into this standard circle format, allowing easy identification of its characteristics.

Graph Sketching

Graph sketching involves drawing a visual representation of an equation on a coordinate plane. It's a crucial skill for interpreting mathematical relationships and understanding the geometric nature of equations. When sketching the graph of a circle, the initial steps involve identifying the circle's center and radius from its equation.
For the given problem, once you determine the center \( (1, 1)\) and the radius \( \sqrt{2}\), place a dot at the center in the coordinate plane.
Then, using the radius, plot points around the center by measuring a distance \( \sqrt{2}\) away in all directions.
Finally, connect these points smoothly to form the circle. Usually, using a compass or circular template ensures precision.

• Begin with plotting the center.
• Measure the radius accurately in horizontal, vertical, and diagonal directions.
• Draw the outer frame of the circle, maintaining an even distance from the center.

This visual approach helps in better understanding spatial relationships, crucial for fields like physics and engineering.

Trigonometric Conversion

Trigonometric conversion is the process of using trigonometric identities and formulas to translate between different mathematical expressions, often between polar and Cartesian forms.
When dealing with polar equations, the main functions you use are sine (\( \sin \theta\)) and cosine (\( \cos \theta\)). These two ratios form the foundation for converting polar coordinates (\((r, \theta)\)) into Cartesian coordinates (\((x, y)\)).
Consider the fundamental identities:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r^2 = x^2 + y^2\)

These relations facilitate smooth transition between forms, allowing easier analysis of the given equation. Decomposing a polar equation using these identities not only assists in plotting graphs but also simplifies complex calculations into more manageable algebraic manipulation.