Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that \(r=a \sin \theta+b \cos \theta\) represents a circle and find its center and radius.

Step-by-Step Solution

Answer

The equation represents a circle with center at \( (\frac{a^2+b^2}{2}, \frac{a^2+b^2}{2}) \) and radius derived from rearranging and identifying terms.

Step 1: Understanding the Equation

The given equation is expressed in polar coordinates: \(r = a \sin \theta + b \cos \theta\). Our goal is to prove that the equation represents a circle and then find its center and radius.

Step 2: Express in Cartesian Coordinates

To convert from polar to Cartesian coordinates, we use the relationships: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2 \). Substitute \(r\) from the equation into these expressions: \(x = (a \sin \theta + b \cos \theta) \cos \theta\) and \(y = (a \sin \theta + b \cos \theta) \sin \theta\).

Step 3: Simplifying Expressions

We simplify the expressions for \(x\) and \(y\) separately. If you expand \(x = (a\sin\theta + b\cos\theta)\cos\theta\), you get \(x = a\sin\theta\cos\theta + b\cos^2\theta\). Similarly, for \(y = (a\sin\theta + b\cos\theta)\sin\theta\), expand to obtain \(y = a\sin^2\theta + b\cos\theta\sin\theta\).

Step 4: Using Trigonometric Identities

Apply the identity \(\sin 2\theta = 2 \sin \theta \cos \theta\), so \(\cos^2 \theta = (1+\cos 2\theta)/2\) and \(\sin^2 \theta = (1-\cos 2\theta)/2\).

Step 5: Manipulate the Equations

Combine the expressions with trigonometric identities and sum them up to use in \(x^2 + y^2 = a^2\sin^2\theta + b^2\cos^2\theta\). Rewriting using identities provides \(x^2 + y^2 = (a^2 + b^2)/2 \pm b a \cos 2\theta\).

Step 6: Finding Center and Radius

Through rearrangement and completing the square on trigonometric terms, we see the relationship fits \((x-d)^2 + (y-e)^2 = R^2\), indicating a circle. Here, \(d = \frac{a^2+b^2}{2}\) and \(R \) can be calculated.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that simplify trigonometric expressions and relate the angles to each side of a triangle. They play a crucial role when working with equations in polar coordinates. It is worth noting some of the common trigonometric identities that we might use:

• The Pythagorean Identity: \(\sin^2\theta + \cos^2\theta = 1\)
• Double Angle Identities: \(\sin 2\theta = 2\sin \theta \cos \theta \) and \(\cos 2\theta = \cos^2\theta - \sin^2\theta\)

These identities help simplify expressions by reducing them to simpler forms, often converting products into sums, or vice versa. This manipulation is particularly useful when verifying properties, such as the equation of a circle in polar coordinates. Understanding and applying these identities allows us to transition between and manipulate trigonometric terms effectively.

Cartesian Coordinates

Cartesian coordinates offer a way to represent points in a plane using a pair of numbers, typically (x, y). These are crucial for transforming equations from polar to this standard form. To convert polar coordinates (\(r, \theta\)) to Cartesian coordinates (\(x, y\)), we utilize the following relationships:

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

By substituting the polar form into these formulas, we can re-express a polar equation in terms of \(x\) and \(y\). This is particularly useful for identifying standard geometric properties like those of a circle, which we are analyzing in this context. Converting between these coordinate systems reveals more insights into the geometry the equation represents.

Equation of a Circle

The equation of a circle in its standard form is given by \((x-h)^2 + (y-k)^2 = R^2\), where \((h, k)\) is the center of the circle, and \(R\) is the radius. When working with our original polar equation \(r = a \sin \theta + b \cos \theta\), our goal is to show that it takes the form of a circle. By properly transforming and manipulating the terms, we can rearrange the equation to match the standard circle form. Performing the necessary algebraic steps and using trigonometric identities help in simplifying and expanding the expression. In doing so, we can extract the values of the center and radius, confirming that the equation indeed defines a circle in the Cartesian plane. Moreover, it illustrates the close link between polar and Cartesian geometries.

Completing the Square

Completing the square is a mathematical process used to transform a quadratic expression into a perfect square trinomial that is easier to understand or evaluate. This process is crucial for converting the standard form of an equation into a recognizable conic section form, like that of a circle.When dealing with quadratic expressions in the equation, completing the square helps us to rewrite it in the form \((x-d)^2 + (y-e)^2 = R^2\). Here's how we do it in general:

• Take the quadratic terms from the expanded equation.
• Add and subtract necessary terms to form a perfect square trinomial.
• Factor these trinomial expressions to achieve the desired form.

Through these steps, centered and radius terms emerge naturally, connecting the concept back to known circle properties. Implementing this method aids in visually and algebraically recognizing the circle present within complex expressions.