Problem 47 - Algebra and Trigonometry

Question

47–50 Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $$\left(x^{2}+y^{2}\right)^{3}=4 x^{2} y^{2}$$

Step-by-Step Solution

Answer

Convert to polar form, then sketch circles using \(r = \pm \sin 2\theta\).

Step 1: Recognize the Equation Components

The equation given is \((x^2 + y^2)^3 = 4x^2y^2\). Notice that \(x^2 + y^2\) is the expression for \(r^2\) in polar coordinates, and both \(x\) and \(y\) can be expressed in polar form.

Step 2: Convert to Polar Coordinates

To convert the rectangular equation to polar form, substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the equation. The left-hand side becomes \((r^2)^3 = r^6\). For the right-hand side, replace \(x^2\) with \((r \cos \theta)^2 = r^2 \cos^2 \theta\) and \(y^2\) with \((r \sin \theta)^2 = r^2 \sin^2 \theta\). Thus, the equation becomes:\[r^6 = 4 (r^2 \cos^2 \theta)(r^2 \sin^2 \theta) = 4r^4 \cos^2 \theta \sin^2 \theta\]

Step 3: Simplify and Solve for r

Divide both sides of the equation by \(r^4\) to obtain:\[r^2 = 4 \cos^2 \theta \sin^2 \theta\]Simplify the right-hand side using the identity \(\sin 2\theta = 2 \sin \theta \cos \theta\), so \(\cos^2 \theta \sin^2 \theta = \left(\frac{1}{4}\right) \sin^2 2\theta\). Thus, the equation becomes:\[r^2 = 4 \left(\frac{1}{4}\right) \sin^2 2\theta = \sin^2 2\theta\]

Step 4: Solve for r and Interpret Graphically

Taking the square root of both sides gives \(r = \pm \sin 2\theta\). Since \(r\) can be positive or negative, the graph represents circles with their centers at the origin. The graph is symmetric with respect to both the x-axis and the y-axis, forming what appear to be petals around the origin.

Key Concepts

Rectangular Equation

Rectangular equations are the traditional way of describing curves and shapes in the Cartesian coordinate system, using variables such as \(x\) and \(y\). In the rectangular equation \[(x^2 + y^2)^3 = 4x^2y^2,\]the expressions involve powers and products of \(x\) and \(y\). This form of equation is most used in algebra and calculus, where we often analyze relationships involving distances and alignments in a two-dimensional plane. To sketch graphs or simplify calculations, especially in complex situations, converting a rectangular equation to polar coordinates might be beneficial, as it can simplify the equation by taking advantage of the geometric properties involved. Polar coordinates express the position of points in terms of distance \(r\) from a reference point and an angle \(\theta\) from a reference direction.

Radian Measure

Radian measure is a crucial concept in trigonometry and calculus, particularly when working with polar coordinates. Unlike degrees, radians provide a natural and algebraically convenient way to represent angles. The radian measure of an angle is based on the radius of the circle. One radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of that circle.

• There are \(2\pi\) radians in a complete circle.
• This measure allows for direct connection with arc lengths and circumferences.

In the context of converting equations to polar form, understanding radian measure helps in dealing precisely with trigonometric functions like sine and cosine, which depend on angles measured in radians. Using radians, the conversion formula relies on trigonometric identities, ensuring the smooth transformation of equations between coordinate systems.

Trigonometric Identities

Trigonometric identities are essential tools for simplifying and solving equations involving trigonometric functions. They allow us to express these functions in various equivalent forms, thereby facilitating algebraic manipulations. When dealing with polar coordinates and conversion from rectangular forms, trigonometric identities such as:

• \(\sin^2\theta + \cos^2\theta = 1\)
• \(\sin 2\theta = 2\sin\theta \cos\theta\)
• \(\cos 2\theta = \cos^2\theta - \sin^2\theta\)

are especially useful.In the context of the given exercise, these identities help in simplifying the expression \(4 \cos^2 \theta \sin^2 \theta\) to a more manageable form, such as \(\sin^2 2\theta/4\). By applying these identities, one can derive symmetric properties of graphs, such as circular or spiral structures, from what initially appears to be a complex algebraic equation. They aid in understanding the nature of the graph by revealing inherent symmetries and patterns inherent in polar expressions.