Problem 35 - Calculus Early Transcendentals

Question

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=8 \sin \theta\)

Step-by-Step Solution

Answer

Question: Convert the polar equation \(r = 8\sin\theta\) into Cartesian coordinates and describe the resulting curve. Answer: The Cartesian equation is \(x^2 + (y-4)^2 = 16\), which represents a circle centered at (0, 4) with a radius of 4.

Step 1: Replace r with equivalent Cartesian expression

Given the equation \(r = 8\sin\theta\), we want to replace the \(r\) with its equivalent Cartesian expression. To do this, we can use the Pythagorean theorem, which states that \(r^2 = x^2 + y^2\). In this case, we have \(r = \sqrt{x^2 + y^2}\). So, we will replace \(r\) in the original equation with this expression: \((\sqrt{x^2 + y^2}) = 8\sin\theta\) Now, we need to eliminate the \(\sin\theta\) term in order to have an equation in Cartesian coordinates.

Step 2: Replace \(\sin\theta\) with equivalent Cartesian expression

To replace \(\sin\theta\), we can divide the Cartesian equation for \(y\) (i.e., \(y = r\sin\theta\)) by the Cartesian equation for \(r\) (i.e., \(r = \sqrt{x^2 + y^2}\)). Doing so, we get: \(\frac{y}{\sqrt{x^2 + y^2}} = \sin\theta\) Now we can eliminate the \(\sin\theta\) term in the original equation by substituting the expression we just found: \(\sqrt{x^2 + y^2} = 8\left(\frac{y}{\sqrt{x^2 + y^2}}\right)\)

Step 3: Simplify the equation

Now that we have an equation with only Cartesian coordinates, we can simplify it. First, we'll multiply both sides by \(\sqrt{x^2 + y^2}\) to get rid of the fraction: \((\sqrt{x^2 + y^2})^2 = 8y\) Squaring the left side, we get: \(x^2 + y^2 = 8y\)

Step 4: Describe the resulting curve

To describe the curve represented by the equation \(x^2 + y^2 = 8y\), we can try rewriting it in the standard form of a conic section. To do this, we'll complete the square for the \(y\)-term: \(x^2 + (y-4)^2 = 4^2\) This equation represents a circle centered at \((0, 4)\) with a radius of 4. Therefore, the curve described by the given polar equation in Cartesian coordinates is a circle centered at \((0, 4)\) with a radius of 4.

Key Concepts

polar coordinates

Polar coordinates provide an alternative way to describe the position of a point in a plane. Instead of using the traditional

• Cartesian coordinates (where we describe points using an \((x, y)\) pair),
• polar coordinates define a point by two values: \(r\) and \(\theta\).

Here, \(r\) represents the distance from the point to the origin (the center of the coordinate system), and \(\theta\) is the angle that the line from the origin to the point makes with the positive x-axis.Polar coordinates are useful in scenarios involving circles and angles, such as problems in physics and engineering. They provide a more natural description for rotational movements and radial distances.
When you need to switch between polar and Cartesian coordinates, certain relationships help with conversion:- \(x = r \cos \theta\)- \(y = r \sin \theta\)- \(r = \sqrt{x^2 + y^2}\)- \(\tan \theta = \frac{y}{x}\)

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that relates to the lengths of sides in a right triangle. It states that in a right triangle:\[a^2 + b^2 = c^2\]

• where \(c\) is the length of the hypotenuse
• and \(a\) and \(b\) are the lengths of the other two sides.

This theorem is not only limited to triangles. It finds its application in converting polar coordinates to Cartesian coordinates, as seen in our exercise. By letting \(x = a\) and \(y = b\), the equation becomes \[x^2 + y^2 = r^2\].This formula is pivotal when transforming equations from polar to Cartesian forms. For instance, in the present problem, we replace \(r\) with \(\sqrt{x^2 + y^2}\). This allows us to express everything in terms of \(x\) and \(y\).
Whenever you need to find distances or work with equations involving circles, the Pythagorean theorem often comes handy.

conic sections

Conic sections are curves obtained by intersecting a plane with a cone. These sections include a major group of shapes such as circles, ellipses, parabolas, and hyperbolas. The properties and equations of these figures are fundamental in mathematics and science.

• A circle is a set of points in a plane that are equidistant from a given point (the center).
• An ellipse is similar but stretched along either horizontal or vertical directions.
• Parabolas are U-shaped curves important in optics and physics.
• Hyperbolas have two branches and appear in advanced mathematics and science.

For our problem, the equation \(x^2 + y^2 = 8y\) represents a conic section called the circle. By rewriting it to its standard form, we see that this circle has a specific location and size, centered at \((0, 4)\) with radius 4.
Understanding conics is crucial, as they appear unexpectedly in diverse fields like astronomy, physics, and engineering.

circle equation

The equation of a circle in its standard Cartesian form is:\[(x - h)^2 + (y - k)^2 = r^2\]Here,

• \((h, k)\) is the center of the circle
• \(r\) is the radius of the circle.

Whenever you encounter an equation like \(x^2 + y^2 = 8y\), you may recognize parts of it but need to rearrange it into the standard form. In this exercise, by completing the square for the \(y\)-terms, we achieve:\[x^2 + (y - 4)^2 = 16\]This tells us a lot about the circle:

• The center is at point \((0, 4)\).
• The radius is \(4\) (since \(4^2 = 16\)).

Understanding and using the equation of a circle is beneficial in geometry and practical situations, ranging from design to navigation.