Problem 66 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r^{2}=\sin 2 \theta$$

Step-by-Step Solution

Answer

The rectangular coordinates equation is \\( (x^2 + y^2)^2 = 2xy \\).

Step 1: Understanding Polar and Rectangular Coordinates

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle. In rectangular coordinates, the point is represented as \( (x, y) \). We need to express \(r^2 = \sin 2\theta\) in terms of \(x\) and \(y\).

Step 2: Using Polar to Rectangular Conversion Formulas

Recall the conversion formulas: \(x = r\cos\theta\) and \(y = r\sin\theta\). Also, \(r^2 = x^2 + y^2\) can be used to express \(r^2\) in terms of \(x\) and \(y\).

Step 3: Expressing Sine Double Angle Identity

Know that the sine double angle identity states \(\sin 2\theta = 2\sin \theta\cos\theta\). We can substitute this in our equation to get \(r^2 = 2\sin\theta\cos\theta\).

Step 4: Substitute the Conversion Equations

From the polar to rectangular formulas, substitute \(\sin\theta = \frac{y}{r}\) and \(\cos\theta = \frac{x}{r}\) into the double angle identity: \(2\sin\theta\cos\theta = 2\left(\frac{y}{r}\right)\left(\frac{x}{r}\right) = \frac{2xy}{r^2}\).

Step 5: Solve for Rectangular Equation

Substitute \(r^2\) into our equation: \(r^2 = \frac{2xy}{r^2}\). By cross-multiplying, we get \((r^2)^2 = 2xy\). Since \(r^2 = x^2 + y^2\), replacing gives us \((x^2 + y^2)^2 = 2xy\).

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way of describing a point's position in a plane. Instead of relying on two perpendicular axes like the traditional rectangular system, polar coordinates employ a radius and an angle. The radius, denoted as \(r\), measures the distance from the origin to the point. The angle, given as \(\theta\), represents the counterclockwise rotation from the positive x-axis to the radius line. This system works effectively for problems involving circular and rotational symmetry.

Consider the point \((r, \theta)\) in polar coordinates that may represent many different combinations in rectangular coordinates, depending on the angle and radius. For example, points like the edge of a fan blade or any location on a circular path would be efficient to express using polar notation. As vectors can also be described via polar coordinates, this system is essential in fields that study waveforms, oscillations, and other cyclical phenomena.

Remember, converting from polar to rectangular coordinates involves understanding the interplay between these two systems, which is facilitated by the conversion formulas and identities like those used in the given exercise.

Rectangular Coordinates

Rectangular coordinates, usually expressed as \((x, y)\), are familiar from school geometry. They locate a point by plotting a distance along the x-axis (horizontal) and y-axis (vertical). This method visually maps broad ranges of functions and shapes due to its straightforward grid-like structure.

In the context of conversion from polar, each point in rectangular coordinates connects to its polar counterpart. The equation for converting is straightforward:

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

These relationships emanate from trigonometric principles applied to right-angled triangles formed by \(r\), \(\theta\), and the x and y components. Understanding and using these formulas provides a seamless translation between coordinate systems, crucial for applications in physics and engineering fields. The exercise showcases this in expressing an entire polar equation within the comfortable structure of rectangular equations.

Sine Double Angle Identity

Trigonometric identities are formulas that allow manipulation and simplification of complex equations, with the sine double angle identity being particularly useful. This identity, expressed as \(\sin 2\theta = 2\sin \theta\cos \theta\), provides a valuable tool in transforming polar expressions.

In solving the exercise, the identity helps in breaking down \(\sin 2\theta\) into products of \(\sin \theta\) and \(\cos \theta\), which ties back to our conversion formulas. This allows a direct association with the rectangular components \(x\) and \(y\) once expressed as \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\).

By employing this identity and substituting these expressions, we derive a more manageable form in rectangular coordinates, showcasing how identities bridge different mathematical concepts — from angles and trigonometry to spatial and algebraic expressions.