Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-2 a y=0$$

Step-by-Step Solution

Answer

Therefore, the polar form of the rectangular equation \(x^{2}+y^{2}-2 a y=0\) is \(r = 2a\sin(\theta)\).

Step 1: Write Down the Polar Form

The polar form can be illustrated as: \(r = \sqrt{x^{2}+y^{2}}\), \(x = r\cos(\theta)\), \(y = r\sin(\theta)\).

Step 2: Substitute x and y in terms of r and theta

Replace \(x\) and \(y\) in our given equation \(x^{2}+y^{2}-2 a y=0\) by their polar counterparts. This yields: \((r\cos(\theta))^{2}+(r\sin(\theta))^{2}-2a(r\sin(\theta))=0\).

Step 3: Simplify the Equation

Now, simplify the equation obtained in Step 2. The left term simplifies via the pythagorean identity into \( r^{2} = 2ar\sin(\theta)\). This simplifies further to \( r = 2a\sin(\theta)\).

Step 4: Express Final Equation in Polar Form

Finally, the polar form of the equation \(x^{2}+y^{2}-2 a y=0\) is \(r = 2a\sin(\theta)\).

Key Concepts

Rectangular to Polar Conversion

The process of converting a rectangular equation to a polar form involves changing the coordinate system from a Cartesian (rectangular) coordinate system to a polar coordinate system. In the rectangular coordinate system, points are located using the coordinates \(x\) and \(y\). These represent horizontal and vertical distances respectively from a fixed reference point called the origin.

The polar coordinate system, however, uses a different approach. Instead of horizontal and vertical distances, it uses a distance \(r\) (radius) from the origin and an angle \(\theta\) from the positive x-axis. This is particularly useful in equations involving circles and other curves, where relationships between distances and angles are more intuitive.

To convert a rectangular equation to polar form:

• Use the formulas \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
  
  
• Replace every \(x\) and \(y\) in the equation with these expressions.
  
  
• Simplify the equation, using known trigonometric identities when possible, to express it in terms of \(r\) and \(\theta\).

After converting, you can often see relationships and properties that were not obvious in rectangular form.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, rooted in the Pythagorean theorem. It states that for any angle \(\theta\), the square of the sine function plus the square of the cosine function equals one:

\[\sin^2(\theta) + \cos^2(\theta) = 1\]

This identity is universal for all angles and plays a crucial role in simplifying and converting trigonometric equations.

In the context of polar and rectangular conversions, the Pythagorean identity helps to simplify expressions involving \(r\). For a point \( (x, y) \) in rectangular coordinates, the equivalent polar expression is \( r = \sqrt{x^2 + y^2} \), derived directly from the Pythagorean identity.

When manipulating an equation, this identity can also help to eliminate terms and simplify the relationship between variables, making it easier to isolate \(r\) or \(\theta\), as we simplified the step \( r^2 = x^2 + y^2 \) to simply \( r = 2a \sin(\theta) \) in our example.

This simplification demonstrates how powerful the Pythagorean identity can be, providing an elegant way to clean up complex expressions.

Trigonometric Substitution

Trigonometric substitution is a method used to replace algebraic expressions with trigonometric functions, facilitating easier manipulation of the equation. This is especially handy in calculus and when dealing with polar coordinates.

This technique is applied by replacing one or more parts of an equation with a trigonometric entity. For example, in converting rectangular to polar forms, \(x\) and \(y\) are replaced by \(r \cos(\theta)\) and \(r \sin(\theta)\) respectively.

By doing this, complex algebraic equations are transformed into simpler trigonometric forms. This simplification becomes apparent in equations that revolve around circles, spirals, and waves.

Some other common substitutions include:

• Substituting \(\tan(\theta)\) for expressions involving \(\frac{y}{x}\) when dealing with slope or angles.
  
• Using \(\sec(\theta)\) and \(\csc(\theta)\) for reciprocal identities.

Trigonometric substitution allows the harnessing of trigonometric identities and their known values to simplify and solve equations, making even daunting equations more approachable.