Problem 19 Sketch the polar curve. $$r=-1... [FREE SOLUTION]

Chapter 10: Problem 19

Sketch the polar curve. $$r=-1+\sin \theta.$$

Short Answer

Expert verified

To sketch the polar curve $r = -1 + \sin\theta$, we find key points like the minimum and maximum values of $r$ and the corresponding $\theta$ values. In this case, the min and max values of $r$ are -2 and 0, with $\theta$ values of $3\pi/2$ and $\pi/2$, respectively. The curve has symmetry about the vertical axis, so we plot the points $(-2, 3\pi/2)$, $(0, \pi/2)$, and $(0, 3\pi/2)$, then draw the curve by connecting these points smoothly and adding the symmetrical half of the curve.

Step by step solution

Determine the range of 胃

The polar curve will be plotted for 胃 ranging from 0 to $2\pi$ as this range will give us a complete view of the curve.

Identify key points

To sketch the curve, find the key points by determining the maximum and minimum values of $r$ and the corresponding values of $\theta$. We have the equation: $r=-1+\sin \theta$ To find the minimum and maximum values of $r$, we should look at the possible values of $\sin\theta$. Since the sine function ranges from -1 to 1, we can find the minimum and maximum values of $r$ by considering the extreme values for $\sin\theta$. Minimum $r$: When $\sin\theta = -1$, $r = -1 + (-1) = -2$ Maximum $r$: When $\sin\theta = 1$, $r = -1 + 1 = 0$ Now, let's find the corresponding values for $\theta$: For minimum $r$, $\sin\theta = -1$. This occurs when $\theta = 3\pi/2$. Hence, the point is: $(r, \theta) = (-2, 3\pi/2)$. For maximum $r$, $\sin\theta = 1$. This occurs when $\theta = \pi/2$. Hence, the point is: $(r, \theta) = (0, \pi/2)$.

Determine symmetry

Observe that in the equation $r = -1 + \sin\theta$, replacing $\theta$ with $(\pi - \theta)$ results in $r = -1 + \sin(\pi - \theta) = -1 + \sin\theta$. Thus, the curve has symmetry about the vertical axis. Also, replacing $\theta$ with $(\pi + \theta)$ results in $r = -1 + \sin(\pi + \theta) = -1 - \sin\theta$. Thus, the curve doesn't have symmetry about the horizontal axis.

Sketch the curve

Using the key points determined in Step 2 and the symmetry properties determined in Step 3, plot the polar curve: 1. Plot the point $(r, \theta) = (-2, 3\pi/2)$ which represents the minimum value of $r = -2$. 2. Plot the point $(r, \theta) = (0, \pi/2)$ which represents the maximum value of $r = 0$. 3. Utilize the fact that the curve has symmetry about the vertical axis to find the points symmetrical to the ones we already have. The symmetrical point to $(0, \pi/2)$ is $(0, 3\pi/2)$. 4. The curve will start at $(0, \pi/2)$ and pass through the point $(-2, 3\pi/2)$ and end at the point $(0, 3\pi/2)$. Due to the vertical axis symmetry, the other half of the curve will be a mirror reflection. 5. Connect the points smoothly and draw the curve. Having plotted these points and drawn the curve, we have successfully sketched the polar curve $r = -1 + \sin\theta$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates

Polar coordinates are an alternative to the traditional Cartesian coordinate system that most students are familiar with. Rather than using x and y to denote a position on a plane, polar coordinates use a radius and an angle. In polar coordinates, a point is represented by a pair $ (r, \theta) $ where $ r $ is the distance from the origin (radius) and $ \theta $ is the angle measured in radians from the positive x-axis (also known as the polar axis).

This system is particularly useful for dealing with problems involving circular or spiral shapes and movements, and when the relationship between two points is easier to express with angles and distances.

Sine Function

The sine function is a fundamental periodic function in mathematics, often notated as $ \sin(\theta) $. It's significant in various areas of math and science, especially in trigonometry. The sine function takes an angle as input and returns the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. When plotted, it forms a smooth wave-like graph that oscillates between -1 and 1. In polar equations, the sine function can influence the radial distance $ r $ from the origin based on the angle $ \theta $ which results in polar curves with interesting properties like loops and petals.

Symmetry in Polar Graphs

Symmetry in polar graphs means that the graph has a repetitive structure that can be divided into identical halves. Recognizing symmetry can greatly simplify the graphing process. Common symmetries include line symmetry and point symmetry. Line symmetry in polar coordinates typically revolves around the vertical or horizontal axis (polar axis or perpendicular to the polar axis), and point symmetry could be around the origin.

Polar graphs may have symmetry about the line $ \theta = \frac{\pi}{2} $ (vertical line symmetry) or $ \theta = 0 $ (horizontal line symmetry). If replacing $ \theta $ with $ -\theta $ or $ \pi - \theta $ does not change the $ r $ value, the graph has line symmetry. When plotting polar equations, using symmetry can reduce the amount of work needed by allowing us to reflect points across the axis of symmetry.

Plotting Polar Equations

To plot a polar equation, we translate an equation like $ r=-1+\sin(\theta) $ into a visual graph. This involves several steps, such as determining key points, establishing ranges for $ \theta $ and $ r $, and noting symmetries.

Key points are values of $ \theta $ where $ r $ is at a minimum or maximum or where the curve intersects with the origin or an axis. After calculating these, we plot these points on polar graph paper, which is made up of concentric circles and lines that radiate outward, representing angles. Plotting additional points between these key loci can help in creating a more accurate curve. Finally, we connect the points in a smooth continuous way to reveal the shape of the polar curve. Having a clear understanding of each step makes sketching polar curves a systematic and easy-to-follow process.

91影视

Sketch the polar curve. $$r=-1+\sin \theta.$$

Short Answer

Step by step solution

Determine the range of 胃

Identify key points

Determine symmetry

Sketch the curve

Key Concepts

Polar Coordinates

Sine Function

Symmetry in Polar Graphs

Plotting Polar Equations

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