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Magazine Chapter 9: Problem 33
Sketch a graph of the polar equation. $$ r=\sqrt{3}-2 \sin \theta $$
Short Answer
Expert verified
The graph is a lima莽on with an inner loop, symmetrical about the horizontal axis.
Step by step solution
01
Understand the Polar Equation
The given polar equation is \( r = \sqrt{3} - 2 \sin \theta \). This equation expresses \( r \), the radial distance, as a function of \( \theta \), the angle.
02
Recognize the Type of Curve
The equation \( r = a - b \sin \theta \) represents a lima莽on. In this case, \( a = \sqrt{3} \) and \( b = 2 \). Because \( b > a \), the lima莽on will have an inner loop.
03
Determine Key Points
To find key points, consider specific values of \( \theta \):- For \( \theta = \frac{\pi}{2} \), \( r = \sqrt{3} - 2(1) = \sqrt{3} - 2 \).- For \( \theta = \frac{3\pi}{2} \), \( r = \sqrt{3} - 2(-1) = \sqrt{3} + 2 \).- For \( \theta = 0 \), \( r = \sqrt{3} \).- For \( \theta = \pi \), \( r = \sqrt{3} \).
04
Find the Theta for Maximum and Minimum R
The maximum radius occurs when \( \sin \theta = -1 \), yielding \( r_{max} = \sqrt{3} + 2 \). The minimum (negative) radius occurs when \( \sin \theta = 1 \), giving \( r_{min} = \sqrt{3} - 2 \).
05
Plot the Key Points on Polar Coordinate System
Plot the points on the polar coordinate plane for the calculated values. Remember that for a negative \( r \) value, plot the equivalent positive value at \( \theta + \pi \).For example, if \( r = \sqrt{3} - 2 \) is negative, plot at \( \theta = \frac{\pi}{2} + \pi = \frac{3\pi}{2} \).
06
Draw the 尝颈尘补莽辞苍 with Inner Loop
Using the calculated points and understanding that \( b > a \), sketch a lima莽on shape that includes an inner loop starting at the origin when \( r = 0 \) (found when \( \sin \theta = \frac{ rac{\sqrt{3}}{2}}{1} \) or similarly calculated). Ensure the sketch reflects symmetry about the horizontal line passing through the pole.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
尝颈尘补莽辞苍
A 尝颈尘补莽辞苍 is a type of polar graph characterized by distinctive loops and heart-like shapes. This curve is usually defined by equations of the form \(r = a \, \pm \, b \sin \theta\) or \(r = a \, \pm \, b \cos \theta\). Depending on the relation between \(a\) and \(b\), the 尝颈尘补莽辞苍 can have different appearances:
- If \(b > a\), the 尝颈尘补莽辞苍 will have an inner loop, as is the case with the given equation \(r = \sqrt{3} - 2 \sin \theta\).
- If \(b = a\), the 尝颈尘补莽辞苍 forms a cardioid, which resembles the shape of a heart.
- If \(b < a\), the 尝颈尘补莽辞苍 will not have a loop, resembling more of a distorted circle.
Polar Coordinates
Unlike traditional Cartesian coordinates that use \(x\) and \(y\) to locate points in a plane, polar coordinates utilize the distance from the origin \(r\) (the radial distance) and the angle \(\theta\) from a reference direction. This system is especially useful for plotting curves like 尝颈尘补莽辞苍s, which naturally have radial symmetry.
- The angle \(\theta\) is often measured in radians, moving counter-clockwise from the positive x-axis direction.
- The radial coordinate \(r\) can be positive or negative; when negative, the point is located in the opposite direction, effectively adding \(\pi\) to \(\theta\).
Trigonometric Functions
Trigonometric functions like sine and cosine play a fundamental role in polar equations, as they influence the radial distance \(r\) for given angles \(\theta\). For the equation \(r = \sqrt{3} - 2 \sin \theta\):
- \(\sin \theta\) oscillates between -1 and 1, affecting \(r\) considerably. When \(\sin \theta = 1\), \(r\) achieves its minimum value, and when \(\sin \theta = -1\), \(r\) reaches its maximum.
- These functions introduce periodic behavior, which is pivotal when graphing the 尝颈尘补莽辞苍 since they determine the curve's oscillatory nature around the radial directions.
