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Chapter 10: Problem 81

Sketching a Polar Graph In Exercises \(81-92,\) sketch a graph of the polar equation. $$r=8$$

Short Answer

Expert verified
The polar graph of the equation \(r = 8\) is a circle with a radius of 8 units, centred at the origin.

Step by step solution

01

Identify the Polar Equation

The given polar equation is \(r = 8\). This indicates that the radius \(r\), or the distance of each point from the origin, is a constant value of 8 regardless of its angle \(\theta\).
02

Sketch the Graph

Plot a point on each axis that is 8 units from the origin. Draw a circle through these points with the origin as the centre. The circle represents the polar graph of the equation \(r = 8\). This is because the distance of any point on the circle from the origin is equal to 8, which satisfies the given equation.

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Key Concepts

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

Polar Equations
Polar equations represent relationships between points in a coordinate system where each point is defined by a radius and an angle. In the polar coordinate system, a point is given as \((r, \theta)\), where \(r\) is the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line connecting the origin with the point.
This system is particularly useful for equations describing curves like circles, spirals, and other shapes that are symmetric about the origin.
  • The equation \(r = 8\) is an example of a simple polar equation.
  • It tells us that for all points on the graph, the distance from the origin will always be \(8\), regardless of \(\theta\).
Polar coordinates can be particularly insightful for understanding elements in natural phenomena, where symmetrical or radial symmetry is involved.
Graphing Polar Coordinates
Graphing polar coordinates involves plotting points based on their distance from the origin and their angle. This approach is different from the Cartesian system, where points are plotted using \(x\) and \(y\) coordinates based on perpendicular axes.
To graph the equation \(r = 8\):
  • Start by plotting points where the radius \(r\) equals 8 for a range of angles.
  • Visualize this by fixing a point at the origin and moving \(8\) units outward at every angle \(\theta\).
  • Because \(r\) is constant, as you plot these points across all angles from 0 to 360 degrees, they form a circle.
The beauty of polar graphs is that they allow for a simpler interpretation of circular or radial shapes, which may be more complex to describe in Cartesian coordinates.
Circle Graphs
In polar coordinates, a simple way to express a circle is by an equation where the radius \(r\) is constant, such as \(r = 8\). This tells us the circle鈥檚 radius from the center point, which is the origin in polar coordinates.
Here's how it works:
  • The equation \(r = 8\) means every point on the circle is exactly 8 units away from the origin.
  • The circle is the set of all points that satisfy this condition, making it a representation of those points at all angles \(\theta\).
Circle graphs in polar coordinates highlight the symmetry and unity of these distances around a central point. This makes them especially useful in scenarios involving waves, rotations, and periodic patterns, where such uniformity is prevalent.

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Most popular questions from this chapter

Finding the Arc Length of a Polar Curve In Exercises \(59-64\) , use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve. $$r=\sec \theta, \quad\left[0, \frac{\pi}{3}\right]$$

Show that the eccentricity of a hyperbola can be written as \(e=\frac{r_{1}+r_{0}}{r_{1}-r_{0}}\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{e+1}{e-1}\)

Finding a Polar Equation In Exercises \(33-38\) , find a polar equation for the conic with its focus at the pole and the given eccentricity and directrix. (For convenience, the equation for the directrix is given in rectangular form.) $$\begin{array}{ll}{\text { Conic }} & {\text { Eccentricity }} \\ {\text { Ellipse }} & {\quad e=\frac{5}{6}}\end{array} \begin{array}{l}{\text { Directrix }} \\ {y=-2}\end{array}$$

Area Find the area of the circle given by \(r=\sin \theta+\cos \theta\) Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

Finding an Angle In Exercises \(107-112,\) use the result of Exercise 106 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\) . Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta .\) Identify the angle \(\psi\) . $$r=2 \cos 3 \theta \quad \theta=\frac{\pi}{4}$$
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