Problem 98 - Precalculus

Question

The use of a different system of coordinates simplifies many mathematical expressions and some calculations are performed in an easier way. The rectangular coordinates \((x, y)\) are transformed into polar coordinates by the equations $$x=r \cos \theta \quad \text { and } \quad y=r \sin \theta$$ where \(r=\sqrt{x^{2}+y^{2}}(r \neq 0)\) and \(\tan \theta=\frac{y}{x}(x \neq 0) .\) In polar coordinates, the equation of the unit circle \(x^{2}+y^{2}=1\) is just \(r=1\) In calculus, we use polar coordinates extensively. Transform the rectangular equation to polar form. $$\left(x^{2}+y^{2}\right)^{2}-16\left(x^{2}-y^{2}\right)=0$$

Step-by-Step Solution

Answer

In polar form, the equation is \(r = 4\sqrt{\cos(2\theta)}\).

Step 1: Understand the given expression

The equation given is \((x^2 + y^2)^2 - 16(x^2 - y^2) = 0\). We need to transform this into polar form using the relationships between rectangular and polar coordinates.

Step 2: Substitute polar coordinate expressions

We know the conversions: \(x = r\cos\theta\) and \(y = r\sin\theta\). Also, \(x^2 + y^2 = r^2\). Substituting these, the equation becomes \((r^2)^2 - 16((r\cos\theta)^2 - (r\sin\theta)^2) = 0\).

Step 3: Simplify the powers of r

Rewrite the expression: \(r^4 - 16(r^2\cos^2\theta - r^2\sin^2\theta) = 0\). Notice that both terms are in terms of \(r^2\), so factor them out.

Step 4: Recall trigonometric identity

The expression \(\cos^2\theta - \sin^2\theta\) is the trigonometric identity for \(\cos(2\theta)\). Substitute this back into the equation.

Step 5: Substitute and factor out common terms

We get \(r^4 - 16r^2\cos(2\theta) = 0\). Factor out \(r^2\): \(r^2(r^2 - 16\cos(2\theta)) = 0\).

Step 6: Set factors to zero and solve

Since \(r eq 0\), the valid solution comes from \(r^2 - 16\cos(2\theta) = 0\). Thus, \(r^2 = 16\cos(2\theta)\). Taking square root gives \(r = 4\sqrt{\cos(2\theta)} \).

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a fundamental part of mathematics used to determine a point's position in a two-dimensional space. This system utilizes two perpendicular lines, the horizontal x-axis and the vertical y-axis, which intersect at a point known as the origin. Each point in this coordinate system is represented as an ordered pair

• First coordinate: x - Position along the horizontal axis.
• Second coordinate: y - Position along the vertical axis.

The beauty of this system is its simplicity for plotting and visualizing points. For example, a point at (3, 4) is located three units to the right of the origin along the x-axis, and four units up along the y-axis.
This coordinate system sets the foundation for more advanced mathematical models and methods, such as transforming these into polar coordinates.

Exploring Trigonometric Identity

Trigonometric identities are essential relationships involving the angles and sides of triangles. One of the most important identities relevant to coordinate transformations is the

• Pythagorean identity: \(\cos^2\theta + \sin^2\theta = 1\)
• And related identities like \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\), instrumental in converting rectangular equations into polar form.

These identities are derived from the unit circle, where the radius is 1, making them handy for calculations in polar coordinates. They simplify expressions that would otherwise require much more complicated algebraic manipulation.
In the context of our problem, the identity \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\) is used to transform and simplify trigonometric expressions when substituting from rectangular to polar equations.

The Art of Coordinate Transformation

Coordinate transformation is the method of converting coordinates from one system to another, allowing for a fresh perspective and simplified expressions. This transformation is pivotal when dealing with curves and complex shapes that are better understood in a different coordinate system.

• Rectangular to Polar: Converting rectangular coordinates \(x, y\) to polar coordinates \(r, \theta\) facilitates easier calculus on curves. This is done using the formulas \(x = r\cos\theta\) and \(y = r\sin\theta\).
• For any point, \(r\) (the radius) is determined as the distance from the origin, using \(r = \sqrt{x^2 + y^2}\), and \(\theta\) (the angle) calculated by \(\tan \theta = \frac{y}{x}\).

By transforming coordinates, problems like our original exercise, which seems complicated in rectangular form, become manageable in polar form, as seen when the unit circle equation simplifies to \(r = 1\).
Such transformations are not just academic exercises but have practical implications in fields such as physics, engineering, and computer graphics, where different perspectives are necessary for solutions.