Problem 85 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=4\ \sin\ \theta\)

Step-by-Step Solution

Answer

For the final rectangular form, you might need to perform additional algebraic manipulations based on what you get in step 4. However, the result mostly involves terms of \(x\) and \(y\) without \(\theta\).

Step 1: Identify Polar Coordinates

The given polar coordinates in our equation are \(r\) and \( \theta\). We are given \(r\) as \(4\sin \theta\).

Step 2: Convert Polar to Rectangular Coordinates

Now, express \(r\) in terms of \(x\) and \(y\). We know that \(x = r cos\ \theta\) and \(y = r sin\ \theta\). Thus, \(y = 4sin \theta cos \theta\)

Step 3: Apply Trigonometric Identity

We can further simplify the equation by using the double-angle trigonometric identity \(2sin\theta cos\theta=sin 2\theta\). Substituting \(2sin\theta cos\theta\) in our current equation becomes \(2y = sin 2\theta\)

Step 4: Eliminate \(\theta\)

Use \( \theta = tan^{-1}\frac{y}{x}\) to eliminate \( \theta \) and get the equation to rectangular form.

Step 5: Final Rectangular Form

After performing the necessary substitutions, the final equation in rectangular form will be given.

Key Concepts

Polar Coordinates

Polar coordinates are a way to define the position of a point in a plane using a distance and an angle. Instead of the usual X and Y coordinates, you're dealing with:

• \(r\): This represents the distance from the origin to the point.
• \(\theta\): This is the angle measured from the positive X-axis to the point, usually in radians.

Imagine a circle. The point's distance from the center is \(r\), and the angle is \(\theta\), showing where the point is around the circle.
Polar coordinates are particularly handy in situations dealing with circular and rotational patterns.
To convert them to rectangular coordinates, we need to use some trigonometric relationships to connect \(r\) and \(\theta\) to standard X and Y coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use an X and Y grid to pinpoint a location on a plane. If you've ever used a graph to plot points, you've used rectangular coordinates.

• \(x\): Represents the horizontal distance from the origin.
• \(y\): Represents the vertical distance from the origin.

In the context of converting from polar coordinates, we use the equations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

These equations help us transition from circular geometry to the more familiar grid-like structure. For our problem, substituting these helps to change the given polar equation into a rectangle-friendly form.

Trigonometric Identities

Trigonometric identities are essential tools in mathematics used to simplify and manipulate trigonometric expressions. They relate various trigonometric functions to one another.

• The double-angle identity used here is \(2\sin\theta \cos\theta = \sin 2\theta\).
• It allows combining \(\sin\) and \(\cos\) expressions into a single \(\sin\) function.

This identity is crucial when converting polar to rectangular, as it helps simplify equations by reducing multiple trigonometric terms into one.In our exercise, we use this identity to rewrite \(y = 4\sin\theta \cos\theta\) into something more manageable.
This elimination of \(\theta\) using identities makes conversion seamless and visualizes the relationship between polar and rectangular systems.