Problem 28 - Algebra and Trigonometry

Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \csc \theta=8 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + (y - 4)^2 = 16 \) which represents a circle with center at \( (0, 4) \) and radius \( 4 \).

Step 1: Rewrite polar equation using trigonometric identity

Start with the given polar equation: \[ r \csc \theta = 8 \] Recall that \( \csc \theta = \frac{1}{\sin \theta} \). Substitute this into the equation: \[ r \cdot \frac{1}{\sin \theta} = 8 \] Simplify to get: \[ \frac{r}{\sin \theta} = 8 \]

Step 2: Use identity for rectangular coordinates

Notice that in polar coordinates, \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). Substitute these into the equation: \[ \frac{r}{\frac{y}{r}} = 8 \] Simplify the expression: \[ \frac{r^2}{y} = 8 \]

Step 3: Simplify to obtain the rectangular form

Since \( r^2 = x^2 + y^2 \), substitute it in the equation: \[ \frac{x^2 + y^2}{y} = 8 \] Multiply both sides by \( y \) to clear the fraction: \[ x^2 + y^2 = 8y \]

Step 4: Rewrite the equation in standard conic form

Rearrange the equation: \[ x^2 + y^2 - 8y = 0 \] To write the equation in standard form, complete the square for the \( y \) term: \[ x^2 + (y^2 - 8y) = 0 \] The term to complete the square is \( (\frac{8}{2})^2 = 16 \), so: \[ x^2 + (y^2 - 8y + 16) = 16 \] This results in: \[ x^2 + (y - 4)^2 = 16 \]

Step 5: Identify and graph the equation

The resulting equation \[ x^2 + (y - 4)^2 = 16 \] is a circle with center at \( (0, 4) \) and radius \( 4 \).

Key Concepts

polar coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle rather than the usual x and y coordinates. They are written as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis. This system is especially useful for dealing with problems involving circular and rotational symmetry. Understanding polar coordinates helps make it easier to convert between different coordinate systems, such as transforming polar equations into rectangular equations.

rectangular coordinates

Rectangular coordinates, or Cartesian coordinates, represent points in a plane using two perpendicular axes: x and y. Each point is defined by its distance from the x-axis and y-axis, written in the form \( (x, y) \). This system forms the basis for most algebra and calculus problems. Converting between polar and rectangular coordinates often involves using trigonometric identities to bridge the relationship between radius, angle, and linear distances.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They are useful for simplifying expressions and solving equations. Common identities include:

• \( \sin \theta = \frac{y}{r} \)
• \( \cos \theta = \frac{x}{r} \)
• \( \tan \theta = \frac{y}{x} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

In polar to rectangular conversions, identities like \( r = \sqrt{x^2 + y^2} \) and \( \tan \theta = \frac{y}{x} \) are frequently used to transform equations and simplify calculations.

graphing conic sections

Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. To graph conic sections:

• Identify the type of equation: for circles, look for equations like \( x^2 + y^2 = r^2 \); for ellipses, look for forms like \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• Rewrite the equation in its standard form by completing the square if necessary.
• Determine key features like center, radius, vertices, and foci.

For example, transforming and graphing \( x^2 + (y - 4)^2 = 16 \) gives a circle centered at \( (0, 4) \) with a radius of 4.