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Magazine Chapter 10: Problem 14
Sketch the polar curve and find polar equations of the tangent lines to the curve at the pole. $$r=4 \sin \theta$$
Short Answer
Expert verified
The tangent lines at the pole are horizontal at \( \theta = \frac{\pi}{2} \).
Step by step solution
01
Understand the Polar Equation
The given polar equation is \( r = 4 \sin \theta \). This describes a limacon with its maximum reaching 4 units. When \(\theta = \frac{\pi}{2}\), \(r = 4\), indicating the maximum distance from the pole.
02
Sketch the Polar Curve
To sketch the curve, plot key points at well-chosen angles such as \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\). At \(\theta = 0\), \(r=0\); at \(\theta = \frac{\pi}{2}\), \(r=4\); at \(\theta = \pi\), \(r=0\); and at \(\theta = \frac{3\pi}{2}\), \(r=-4\), returning to the pole. The curve resembles a heart shape, known as a cardioid, and is symmetric about the vertical axis.
03
Conditions for Tangent Line at the Pole
For \( r = 4 \sin \theta \), the pole occurs when \( r = 0 \). This happens at \( \theta = 0 \) and \( \theta = \pi \). At these points, the radial distance is zero, suggesting the curve meets the pole.
04
Derive Polar Equations for the Tangent Lines
At \(\theta = 0\), the tangent line is horizontal, corresponding to \( \theta = \frac{\pi}{2} \). The equation is \( \theta = \frac{\pi}{2} \). At \(\theta = \pi\), the tangent line is also horizontal, so the line is the same \( \theta = \frac{\pi}{2} \). These lines are horizontal as they represent the curve touching the pole horizontally before returning along its path.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Coordinates
Polar coordinates offer a unique way of describing the location of a point in a plane. In contrast to Cartesian coordinates that use x and y to define a point's distance from the axes, polar coordinates use a radial distance, denoted as \(r\), and an angle, represented by \(\theta\).
Here's how they work:
Here's how they work:
- The radial coordinate \(r\) tells you how far the point is from the origin or pole.
- The angular coordinate \(\theta\) determines the counterclockwise angle from the positive x-axis to the point.
Tangent Lines
A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. It has the same direction as the curve at the point of contact.
In polar coordinates, finding a tangent line involves some additional steps because the curve isn't given in the familiar \(y = f(x)\) form. Instead, we use \(r = g(\theta)\).
When a curve meets the origin, tangent lines can be categorized based on their direction:
In polar coordinates, finding a tangent line involves some additional steps because the curve isn't given in the familiar \(y = f(x)\) form. Instead, we use \(r = g(\theta)\).
When a curve meets the origin, tangent lines can be categorized based on their direction:
- At \(\theta = 0\) or \(\theta = \pi\), the tangent lines can be horizontal or vertical, depending on the curve鈥檚 orientation as it approaches the pole.
- Horizontal tangent lines occur when the line is perpendicular to a radius, appearing flat.
Cardioid
A cardioid is a heart-shaped curve that appears in polar equations and is characterized by its distinct symmetrical shape. The term cardioid comes from the Greek word "kardia," meaning heart.
The polar equation for a cardioid often takes forms such as \(r = a(1 + \sin \theta)\) or \(r = a(1 + \cos \theta)\).
The polar equation for a cardioid often takes forms such as \(r = a(1 + \sin \theta)\) or \(r = a(1 + \cos \theta)\).
- In our example \(r = 4 \sin \theta\), setting \(a = 4\), the maximum radial distance is achieved when \(\theta = \frac{\pi}{2}\).
- This results in a symmetrical heart shape centered around the pole, making it easier to sketch when key points and symmetries are identified.
Limacon
A limacon is a type of polar curve that can present itself in various forms: with an inner loop, a dimple, or a cardiod, depending on the coefficients in its equation. The general form is \(r = a + b \sin \theta\) or \(r = a + b \cos \theta\).
The shape of the limacon is determined by the relative sizes of \(a\) and \(b\):
The shape of the limacon is determined by the relative sizes of \(a\) and \(b\):
- If \( |a| = |b|\), the curve forms a cardioid.
- If \( |a| > |b| \), the limacon has a dimple.
- If \( |a| < |b| \), a loop appears inside the limacon.
