Problem 18 - Calculus Early Transcendentals

Question

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=2 \sin \theta+2 \cos \theta$$

Step-by-Step Solution

Answer

The curve is a circle centered at (1,1) with radius \(\sqrt{2}\).

Step 1: Express sine and cosine in Cartesian coordinates

First, we'll use the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, know that \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\). Substitute \(r=2\sin\theta + 2\cos\theta\) into these relationships.

Step 2: Substitute polar relations into the given equation

Substitute the sine and cosine expressions into the given polar equation. You have: \[r = 2\left(\frac{y}{r}\right) + 2\left(\frac{x}{r}\right)\]Multiply through by \(r\) to clear the fractions:\[r^2 = 2y + 2x\]

Step 3: Express \(r^2\) in terms of \(x\) and \(y\)

Recall the polar to Cartesian conversion where \(r^2 = x^2 + y^2\). Replace \(r^2\) with \(x^2 + y^2\) in the equation:\[x^2 + y^2 = 2x + 2y\]

Step 4: Rearrange into standard form

Rearrange the equation into the standard form by completing the square for \(x\) and \(y\).\[(x^2 - 2x) + (y^2 - 2y) = 0\]

Step 5: Complete the square

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

Step 6: Identify the type of curve

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

Key Concepts

Cartesian coordinates

In mathematics, Cartesian coordinates allow us to pinpoint a location in a two-dimensional plane using two values: usually represented as \((x, y)\). These coordinates describe a point's horizontal and vertical positions from an origin, where the horizontal axis is the x-axis, and the vertical axis is the y-axis.

• Given any point, \(x\) indicates the distance along the x-axis.
• The \(y\) coordinate specifies the distance along the y-axis.

With these two values, we can locate any point precisely in a plane. Cartesian coordinates provide the framework we frequently use in algebra, geometry, and calculus to analyze the positions and movements of points and shapes.

Polar coordinates

Polar coordinates, unlike Cartesian coordinates, describe a point's position based on its distance and direction from a fixed point, known as the origin. Here, a point is represented by \((r, \theta)\), where:

• \(r\) is the radius or distance from the origin.
• \(\theta\) is the angle with respect to the positive x-axis.

This system is particularly useful in scenarios involving rotation around a center point. For instance, when analyzing circular motion or defining curves like spirals and circles, polar coordinates offer more convenience than Cartesian coordinates.To convert between these systems, we utilize relationships such as:

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

These conversions let us transition between the coordinate systems, enriching our analytical toolkit.

Circle equation

A circle is a special kind of curve in a plane, defined by all points equidistant from a center point. In Cartesian coordinates, a circle's equation typically follows the format:\[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) is the center of the circle, and \(r\) is its radius.When the center is located at the origin \((0, 0)\), the equation simplifies to:\[x^2 + y^2 = r^2\]In this exercise, we discovered that our polar equation translated into the Cartesian form representing a circle:\[(x - 1)^2 + (y - 1)^2 = 2\]Here, the circle is centered at \((1, 1)\) with a radius \(\sqrt{2}\). Identifying and manipulating these equations is fundamental in geometry, allowing us to visualize and interpret circular shapes.

Trigonometric identities

Trigonometric identities are fundamental relationships between trig functions that hold true for all angles. They help simplify complex expressions in both pure and applied mathematics, including polar to Cartesian conversions.Two crucial identities we often rely on during conversions are:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

These identities allow us to express trigonometric functions in terms of Cartesian coordinates:

• \(\sin \theta = \frac{y}{r}\)
• \(\cos \theta = \frac{x}{r}\)

By substituting these expressions, we can transform equations between systems, making identifications like the circle equation much clearer. Mastery of these identities simplifies many algebraic processes, enabling deeper insights into mathematical relationships.