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Chapter 10: Problem 37

Sketch the curve in polar coordinates. $$r=-3-4 \sin \theta$$

Short Answer

Expert verified
The curve is a lima莽on with an inner loop, due to \(|b| > |a|\).

Step by step solution

01

Identify the Curve Type

The equation is given in polar coordinates: \( r = -3 - 4 \sin \theta \). This equation is of the form \( r = a + b \sin \theta \), where \( a = -3 \) and \( b = -4 \). This indicates that the curve is a lima莽on (a type of polar equation) with an inner loop because \( |b| > |a| \).
02

Find Key Points

To sketch the curve, evaluate \( r \) at key angles: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).- At \( \theta = 0 \), \( r = -3 - 4 \sin 0 = -3 \).- At \( \theta = \frac{\pi}{2} \), \( r = -3 - 4 \sin \frac{\pi}{2} = -3 - 4 = -7 \).- At \( \theta = \pi \), \( r = -3 - 4 \sin \pi = -3 \).- At \( \theta = \frac{3\pi}{2} \), \( r = -3 - 4 \sin \frac{3\pi}{2} = -3 + 4 = 1 \).
03

Sketch the Inner Loop

Since there is an angle where \( r \) becomes zero, find when the loop ends. Solve \(-3 - 4 \sin \theta = 0\):\[ \sin \theta = -\frac{3}{4} \]This equation shows two angles where \( r = 0 \), indicating the start and end of the inner loop. The inner loop will be drawn in the quadrant where these angle values reside, usually reflecting across the origin for symmetric shapes like lima莽ons.
04

Sketch the Complete Curve

Plot the r-values computed in Step 2 on polar paper. Remember that negative r-values indicate a point in the opposite direction by \(180^\circ\). Connect these points smoothly, considering the symmetry and shape of the lima莽on. Ensure the inner loop part is appropriately sketched based on calculations in Step 3.

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Key Concepts

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

Lima莽on Curve
A lima莽on curve is a fascinating type of mathematical curve that can appear in polar coordinates. This curve owes its name to the French word for 'snail' due to its snail-like appearance, depending on its specific form. A general equation for a lima莽on can be expressed as either \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Whether the lima莽on has a loop or not depends on the values of \( a \) and \( b \).
  • If \( |b| > |a| \), the lima莽on has an inner loop.
  • If \( |b| = |a| \), the lima莽on pushes out to a cardioid shape.
  • If \( |b| < |a| \), it becomes a dimpled or convex lima莽on.
Our specific example, \( r = -3 - 4 \sin \theta \), results in a lima莽on with an inner loop because \( |-4| = 4 \) is greater than \( |-3| = 3 \).
Understanding these conditions helps predict the general shape of the curve, which is crucial for sketching polar curves accurately.
Equation in Polar Form
In polar coordinates, points are defined by a radius \( r \) from the origin and an angle \( \theta \) from the positive x-axis. This is different from Cartesian coordinates, which use x and y coordinates to specify points. An equation in polar form usually involves \( r \) as a function of \( \theta \), and it can represent various curves.
  • The equation \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \) is common for representing different forms of the lima莽on curve.
  • These equations can easily describe curves that have one or more loops or other forms, depending on the parameters \( a \) and \( b \).
In our case, \( r = -3 - 4 \sin \theta \) tells us how the radius changes as the angle \( \theta \) rotates around the origin. Identifying these variations allows one to determine characteristics like loops, dimples, or convex shapes.
Knowing the form of the polar equation is key to anticipating the shape you will sketch.
Sketching Polar Curves
Sketching polar curves might seem complex, but it can be simpler by breaking it down step by step. The task involves plotting points based on properly evaluated \( r \)-values at specific angles \( \theta \). Follow these guidelines:
  • Determine the nature of the curve, such as whether it's a lima莽on and if it contains an inner loop.
  • Identify key points by computing \( r \) at incremental angles like \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
  • Investigate symmetry and key features using additional helpful angles.
For example, at \( \theta = 0 \), \( r = -3 \) in our equation, implying a point in the opposite direction since it's negative. At \( \frac{\pi}{2} \), \( r = -7 \), again pointing further along the opposite direction due to the negative sign. Calculate when \( r \) equals zero to locate where an inner loop begins or ends. Connect plotted points considering symmetry naturally present in polar curves such as lima莽ons.
By doing this, you can conclude with a complete and accurate representation of the polar curve.

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Most popular questions from this chapter

Show that the curve with parametric equations $$x=t^{2}-3 t+5, \quad y=t^{3}+t^{2}-10 t+9$$ intersects itself at the point \((3,1),\) and find equations for the two tangent lines to the curve at the point of intersection.

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(a) Show that the Folium of Descartes \(x^{3}-3 x y+y^{3}=0\) can be expressed in polar coordinates as $$ r=\frac{3 \sin \theta \cos \theta}{\cos ^{3} \theta+\sin ^{3} \theta} $$ (b) Use a CAS to show that the area inside of the loop is \(\frac{3}{2}\) (Figure \(3,1.3 a)\)
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