Problem 101 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r^2=\sin\ 2\theta\)

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r^2=\sin\ 2\theta\) is \(x^2 + y^2 = 2xy\).

Step 1: Identify the basic polar to rectangular conversion formulas

Understand and be familiar with the following conversion formulas between polar and rectangular coordinates: \(x=r\cos\ \theta\), \(y=r\sin\ \theta\) and \(r^2 = x^2 + y^2\). These formulas will be used in the conversion process.

Step 2: Begin the conversion process

The given equation is \(r^2=\sin\ 2\theta\). Start converting this equation into rectangular form by recognizing that \(r^2 = x^2 + y^2\). So, replace \(r^2\) with \(x^2 + y^2\) in the equation.

Step 3: Replace r in the sin expression

The other part to convert is \(\sin\ 2\theta\). For this we use the double-angle identity \(\sin\ 2\theta = 2\sin\theta\cos\theta\), which reduces to \(2 \cdot \frac{y}{r} \cdot \frac{x}{r}\) when replaced by \(y=r\sin\ \theta\) and \(x=r\cos\ \theta\). This simplifies to \(2\frac{xy}{r^2}\).

Step 4: Complete the conversion

Now substitute \(r^2 = x^2 + y^2\) and \(2\frac{xy}{r^2}\) into the original equation to complete the transformation. Hence, the equation \(r^2=\sin\ 2\theta\) becomes \(x^2 + y^2 = 2xy\), which is in rectangular form.

Key Concepts

Polar Coordinates

Polar coordinates provide a way to represent points in a plane using the distance from a reference point and an angle from a reference direction. This system is particularly useful when dealing with systems that have circular or rotational symmetry.
In polar coordinates, a point is defined by two values:

• **Radius** ( "): The distance of the point from the origin (also called the pole).
• **Angle** ( heta"): The angle formed with respect to the positive x-axis.

Polar coordinates use the notation \(r, \ heta\). They are widely used because they can simplify equations and calculations involving circular movements or periodic functions. For example, polar coordinates make it easier to describe a circle with equation \(r = constant\) compared to its rectangular equivalent \(x^2 + y^2 = constant^2\).
To convert equations from polar to rectangular form, understanding these relationships is key as they connect the polar and rectangular coordinate systems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use orthogonal axes to specify the position of a point in a plane. They are the most common way to describe locations, using two perpendicular lines that intersect at an origin.

• **X Coordinate (x)**: Represents the horizontal distance from the origin along the x-axis.
• **Y Coordinate (y)**: Represents the vertical distance from the origin along the y-axis.

In this system, a point is noted as \(x, y\). When we talk about conversion from polar to rectangular coordinates, we use transformation formulas:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(r^2 = x^2 + y^2\)

These equations help translate the circular representation (polar) into the more familiar rectangular form. In our example of converting \(r^2 = \sin 2\theta\) to rectangular coordinates, we utilize these fundamental conversions. These allow us to translate circular or rotational motions into linear representations.

Trigonometric Identities

Trigonometric identities are equations that hold true for all angles and are vital in simplifying and converting between different kinds of equations, such as polar and rectangular forms.
One crucial identity is the double angle identity for sine, which is used in the conversion task:

• \(\sin 2\theta = 2\sin\theta\cos\theta\)

This identity allows us to express \(\sin 2\theta\) in terms of its primary trigonometric components. In the context of our problem, it translates into rectangular coordinates using the relationships \(\sin\theta = \frac{y}{r}\) and \(\cos\theta = \frac{x}{r}\).By plugging these into the identity, we can reframe trigonometric expressions within the polar equation as functions of \(x\) and \(y\). This iterative use of identities simplifies complex expressions and ties back to how polar coordinates relate to their rectangular counterparts, allowing us to finish the conversion efficiently.