Question

Consider the two circles \(r=2 a \sin \theta\) and \(r=2 b \cos \theta\), with \(a\) and \(b\) positive. (a) Find the area of the region inside both circles. (b) Show that the two circles intersect at right angles.

Short answer

The area inside both circles involves trigonometric integration, and they intersect at right angles due to orthogonal tangents.

Step-by-step solution

Understanding the Problem

We are given two circles in polar coordinates: Circle 1 with equation \(r = 2a \sin \theta\) and Circle 2 with equation \(r = 2b \cos \theta\). We need to find the area of the region inside both circles and demonstrate that they intersect at right angles. These circles are related to cardioid shapes.

Identifying Points of Intersection

To find where the circles intersect, set their equations equal to each other: \(2a \sin \theta = 2b \cos \theta\). Simplifying, we find \(\tan \theta = \frac{b}{a}\). Thus, \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). These are the angular directions of the intersection points.

Finding Intersection Coordinates

Substitute \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\) back into either original circle equation to find the radius \(r\). Using Circle 1, \(r = 2a \sin\left(\tan^{-1}\left(\frac{b}{a}\right)\right) = 2b\). Therefore, the coordinates are \((2b, \tan^{-1}(\frac{b}{a}))\).

Setting Up the Intersection Area Integral

The area common to both circles can be found by integrating the difference between the areas under the curves in polar coordinates. Integrate from \(\theta = 0\) to \(\theta = \tan^{-1}(\frac{b}{a})\) for one function and from \(\theta = \pi/2\) to \(\theta = \tan^{-1}(\frac{b}{a}) + \pi/2\) for the second to find the overlapping area.

Calculate the Area

Find the area of one loop, \(A = \int_{0}^{\tan^{-1}(\frac{b}{a})} \frac{1}{2} (2a \sin \theta)^2 \, d\theta\), and \(\int_{\pi/2}^{\tan^{-1}(\frac{b}{a}) + \pi/2} \frac{1}{2} (2b \cos \theta)^2 \, d\theta\), and sum these areas to obtain the total intersecting area. Use trigonometric identities to simplify and solve, resulting in areas involving \(a\) and \(b\).

Considering Orthogonal Intersection

To show intersection at right angles, demonstrate that the tangents' slopes at intersection points are negative reciprocals. For Circle 1, \(\frac{dy}{dx} = \cot\theta\), and for Circle 2, \(\frac{dy}{dx} = -\tan\theta\). At the intersection, these slopes are negative reciprocals (\(\cot(\tan^{-1}(\frac{b}{a})) = \frac{a}{b}\), \(-\tan(\tan^{-1}(\frac{b}{a})) = -\frac{b}{a}\)), confirming the 90-degree angle.

Key concepts

Polar Coordinates and Their Importance

Polar coordinates are a way of representing points on a plane using a distance and an angle. In this system, each point is represented by \(r, \theta\), where \(r\) is the distance from the origin and \(\theta\) is the angle away from the positive x-axis. This is often useful when dealing with curves that are circular or spiral in nature, as the description becomes more intuitive compared to Cartesian coordinates.

In polar coordinates, many problems in mathematics and physics become easier to solve. For example, when describing the path of a point moving around a circle or along a spiral, the use of angles and radii provides a direct means of understanding their motion. It’s essential to become familiar with polar coordinates, especially if you’re working in fields that deal with periodic functions or phenomena in circular motion.

An understanding of how to convert between polar and Cartesian coordinates is also important. This conversion allows one to use the powerful tools of calculus in solving real-world problems. The conversion is done using the equations \(x = r \cos \theta\) and \(y = r \sin \theta\), and vice versa, \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).

• Applications in periodic motion and oscillations.
• Makes solving problems involving angles and rotations intuitive.
• Widely used in physics, engineering, and computer graphics.

Beauty of Cardioid Shapes

Cardioids are a type of shape in the realm of polar coordinates, identified by their heart-like appearance. They are best understood through their mathematical definition, typically represented by equations like \(r = 2a(1 + \cos \theta)\) or \(r = 2a(1 + \sin \theta)\). These equations produce figures that loop back on themselves, creating that distinctive heart-like curve.

These shapes are part of a family known as limaçons, and they can be formed in practical scenarios such as the patterns traced by a point on a rolling circle (an epicycloid) or in acoustics to show sound wave patterns. Learning about cardioid shapes extends our comprehension of symmetry, resonance, and amplification patterns found in nature and technology.

In this exercise, we might not directly have cardioid equations, but the two given circles in polar form hint at the cardioid types when considered collectively and contextually. Such shapes are instrumental in displaying the underlying beauty of mathematical synchronicities and the interconnectedness of geometry.

• Helps in understanding patterns in acoustics and optics.
• Inspire various designs in art and architecture.
• Revise seeing polar graphs as beautiful geometric figures.

Trigonometric Identities in Problem Solving

Trigonometric identities form the backbone of solving many mathematical problems, primarily when dealing with angles and rotations. These identities allow us to simplify complex expressions and compute values that are initially not apparent.

In the stated exercise, using identities such as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) is critical. Simplifying the expressions to find intersections or calculating areas typically involves knowing these identities. It's crucial for a student to practice using identities to build confidence in manipulating them to fit the necessary format for solving a problem.

Moreover, these not only help in math exercises but also play pivotal roles in various fields, such as engineering for signal processing, physics in wave mechanics, and anywhere periodic functions are analyzed.

• Handle intricate calculations with ease.
• Useful in diverse applications like statistics and physics.
• Essential for deriving further math functions and shortcuts.

The Concept of Tangent Slopes and Their Usages

A tangent slope in mathematics refers to the gradient of a line tangent to a curve at a given point. This concept is crucial for understanding the behavior and properties of curves without directly computing their entire lengths. It provides insights into the angles of intersection, the direction of movement, and changes in functions.

In the exercise above, examining tangent slopes helps demonstrate that two polar coordinate circles intersect at right angles. Checking the slopes at intersection points to see if they are negative reciprocals allows us to mathematically affirm the orthogonality (90-degree intersection) of these curves.

Recognizing tangent slopes has several real-world applications, such as in road design (ensuring safety through appropriate banking curves), designing robot paths (smooth changes in direction), and in economics (analyzing changes in supply and demand curves). Understanding nd applying tangent slopes make calculus a practical tool rather than a theoretical construct.

• Help confirm orthogonality in geometrical constructions.
• Crucial for calculating certain physics phenomena.
• Facilitates transition from static functions to dynamic analyses.