Problem 70 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$ r=-4 \sin \theta $$

Step-by-Step Solution

Answer

The rectangular form of the given polar equation is \(y=-4+4\left(\frac{x}{\sqrt{x^2+y^2}}\right)^2\). The graph of this equation is a semi-circle running through the origin with a radius of 4, and the bottom-most point is at \((0,-4)\).

Step 1: Convert Polar to Rectangular Form

Starting with the polar equation \(r=-4 \sin \theta\), we need to convert it to rectangular form. Given the formula for y in polar coordinates is \(y=r\sin\theta\), we can directly substitute the polar equation into the formula, giving us the rectangular equation \(y=-4\sin^2\theta\). Now, we need to simplify this equation by using the identity \(\sin^2\theta=1-\cos^2\theta\). Substituting this identity, our equation becomes \(y=-4(1-\cos^2\theta)\).

Step 2: Simplify the Rectangular Equation

Now simplify \(y=-4(1-\cos^2\theta)\) further by expanding the equation. We obtain the equation \(y=-4+4\cos^2\theta\). We replace \(cos\theta\) using the identity \(cos\theta=x/r\), where \(r= \sqrt{x^2+y^2}\). This gives us the equation \(y=-4+4\left(\frac{x}{\sqrt{x^2+y^2}}\right)^2\). This equation cannot be simplified any further.

Step 3: Graph the Rectangular Equation

Our final equation is \(y=-4+4\left(\frac{x}{\sqrt{x^2+y^2}}\right)^2\). To graph this equation, note that for \(x>0\), \(y\) varies from -4 to 0, and for \(x<0\), \(y\) also varies from -4 to 0. The function looks like a U-shaped plot with its bottom at -4. However, because \(r=-4 \sin \theta\) is a circle that goes through the origin, the graph of our function is a semi-circle with radius 4 running through the origin and the bottom-most point at \((0,-4)\).

Key Concepts

Polar Coordinates

Polar coordinates provide a different approach from the more common Cartesian or rectangular coordinate system for denoting the position of a point in a plane. A point's location is determined by its distance from a reference point, known as the pole (similar to the origin in the Cartesian system), and the angle relative to a reference direction, usually the positive x-axis.

In mathematical terms, a point in polar coordinates is represented as \( r, \theta \), where \( r \) is the radial distance from the pole, and \( \theta \) is the polar angle. This system can be particularly useful when dealing with problems involving symmetry about a point, such as circles or spirals.

Rectangular Coordinates

In contrast to polar coordinates, rectangular coordinates are what most students are familiar with. This system uses two perpendicular axes, typically labeled as the x-axis (horizontal) and y-axis (vertical), to define the position of a point. Here, the location of a point is given by two numbers, \( x, y \), which represent the horizontal and vertical distances from the origin, respectively.

The conversion from polar to rectangular coordinates involves trigonometric relationships, which relate the radial and angular components of polar coordinates to the horizontal and vertical components of rectangular coordinates.

Trigonometric Identities

Trigonometric identities are invaluable tools in simplifying expressions involving trigonometric functions. They are equations that are true for all values of the variables involved. One of the foundational identities is the Pythagorean identity, \( \sin^2\theta + \cos^2\theta = 1 \), which is frequently used in converting polar equations to rectangular form, as seen in the step by step solution.

Understanding and applying these identities can turn a complex polar equation into a more manageable rectangular equation, which can then be graphed or further analyzed using algebraic techniques.

Graphing Equations

Graphing equations is a way of visually representing mathematical functions or relationships. When converting polar equations to rectangular form, the graphing process starts to look familiar. However, the shapes can be quite different due to the way polar coordinates define points. For instance, a simple circular shape in polar coordinates may become a more complex equation in rectangular form.

The final step in understanding a polar to rectangular conversion is to graph the obtained rectangular equation. This visual representation can help students spot errors in their conversion process and gain insight into how the same relationship can be expressed and interpreted in different coordinate systems.