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Magazine Chapter 8: Problem 647
Graph \(r=3-5 \sin \theta\).
Short Answer
Expert verified
The graph is a lima莽on with an inner loop.
Step by step solution
01
Understand the Polar Equation
The equation given is a polar equation, where "\( r \)" represents the radius and "\( \theta \)" represents the angle in radians. The equation is \( r = 3 - 5 \sin \theta \). In polar coordinates, this means for each value of \( \theta \), we calculate \( r \) and plot the point \((r,\theta)\) on the polar grid.
02
Identify the Shape of the Graph
The equation has a form \( r = a + b \sin \theta \), which is typical for a lima莽on. Depending on the values of \( a \) and \( b \), lima莽ons can have different shapes (e.g., inner loop, dimpled, or convex). In our case, \( a = 3 \) and \( b = -5 \), and since \(|b| > |a|\), the graph will have an inner loop.
03
Determine Key Points
Calculate specific values of \( r \) for notable angles \( \theta \) (e.g., \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)) to understand the graph better. Substitute these values into the equation:- \(\theta = 0 \rightarrow r = 3 - 5 \times 0 = 3\)- \(\theta = \frac{\pi}{2} \rightarrow r = 3 - 5 \times 1 = -2\)- \(\theta = \pi \rightarrow r = 3 - 5 \times 0 = 3\)- \(\theta = \frac{3\pi}{2} \rightarrow r = 3 - 5 \times (-1) = 8\)- \(\theta = 2\pi \rightarrow r = 3 - 5 \times 0 = 3\)
04
Sketch the Polar Graph
Use the calculated points and connect them smoothly, maintaining the lima莽on shape with an inner loop. Begin at \( \theta = 0\) and trace the curve counterclockwise. At \( \theta = \frac{\pi}{2} \), the radius becomes negative, indicating that the curve crosses the origin and continues in the opposite direction for that angle. A complete revolution \( 0 \leq \theta < 2\pi \) will display the full inner loop.
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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 to plot points on a plane. Unlike the Cartesian system that uses \(x\) and \(y\) coordinates, polar coordinates use a radius and an angle, given as \((r, \theta)\). Here:
- \( r \) represents the distance from the origin to the point on the plane.
- \( \theta \) is the angle measured from the positive x-axis, moving counterclockwise.
Polar Equations
In polar coordinates, equations are expressed with \( r \) and \( \theta \). For example, the polar equation \( r = 3 - 5 \sin \theta \) describes how the radius \( r \) changes with different angles \( \theta \). Such equations can represent complex curves.To understand these equations, note how \( r \) changes as \( \theta \) varies across its range (from 0 to \(2\pi\)). It can provide curves that vary widely depending on the trigonometric functions involved and their coefficients. This specific type of equation often results in distinctive shapes, such as a lima莽on in our case.
Graphing Techniques
Graphing polar equations like lima莽ons involves several strategic steps.
- First, calculate \( r \) for critical angles like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\).
- Plot these points on the polar grid.
- Note any symmetry, which often aids in sketching the rest of the curve.
Trigonometric Functions
Trigonometric functions like sine and cosine play a crucial role in shaping polar graphs. In our equation \( r = 3 - 5 \sin \theta \), the sine function modulates the distance, \( r\), as \( \theta \) changes. Here are some essential insights:
- The function \( \sin \theta \) varies between \(-1\) and \(1\).
- This range affects \( r's \) value, adding complexity to the graph.
- The amplitude and sign of \( b \) (in \( b \sin \theta \)) dictate the extent and direction of this modulation.
