Problem 46 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ x^{2}-y^{2}=1 $$

Step-by-Step Solution

Answer

The polar form is \(r = \pm \sqrt{\frac{1}{\cos(2\theta)}}\).

Step 1: Identify the Polar Coordinates

Recall that in polar coordinates \((r, \theta)\), the relationship between polar and Cartesian coordinates is given by the equations: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). We'll use these formulas to express the given equation in polar form.

Step 2: Substitute Polar Relations

Substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) into the equation \(x^2 - y^2 = 1\). This gives us:\[(r \cos(\theta))^2 - (r \sin(\theta))^2 = 1\] Simplify to:\[r^2 \cos^2(\theta) - r^2 \sin^2(\theta) = 1\]

Step 3: Factor out the Common Factor

Notice that both terms on the left-hand side share a common factor of \(r^2\). Factor this out:\[r^2 (\cos^2(\theta) - \sin^2(\theta)) = 1\]

Step 4: Use Trigonometric Identity

Recall the trigonometric identity \(\cos^2(\theta) - \sin^2(\theta) = \cos(2\theta)\). Substitute this into the equation:\[r^2 \cos(2\theta) = 1\]

Step 5: Solve for r

To express \(r\) in terms of \(\theta\), solve for \(r^2\):\[r^2 = \frac{1}{\cos(2\theta)}\]Take the square root of both sides to find \(r\):\[r = \pm \sqrt{\frac{1}{\cos(2\theta)}}\] This is the polar form of the equation.

Key Concepts

Cartesian coordinates

Cartesian coordinates are a way of representing points in a space using two (or more) numbers. In a two-dimensional plane, these numbers are usually denoted as \(x\) and \(y\). Each point on this plane can be uniquely located using an \(x\) value and a \(y\) value. This \(x\) value tells us how far to move horizontally (right if positive, left if negative) from the origin, whereas \(y\) tells us this same type of movement in the vertical direction (up if positive, down if negative).

When dealing with equations that use Cartesian coordinates, such as \(x^2 - y^2 = 1\), we describe shapes or paths on a flat plane. The given equation is described in Cartesian form, specifying a specific relationship between any \(x\) and any \(y\) such that their squared difference is always 1.

Understanding how these points relate to each other visually on a graph is key to mastering Cartesian coordinates. They are fundamental in graphing equations and are the usual starting point before delving into other coordinate systems like polar coordinates.

trigonometric identities

Trigonometric identities play a crucial role in mathematics, especially when converting between Cartesian and polar coordinates. Simply put, these identities are equations that are always true for any angle \(\theta\). One invaluable identity used in the polar conversion process is \(\cos^2(\theta) - \sin^2(\theta) = \cos(2\theta)\).

These identities are derived from the fundamental trigonometric functions: sine (\(\sin\)) and cosine (\(\cos\)). Here are a few key trigonometric identities:

• Pythagorean Identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• Double Angle Formulas: \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)
• Sum and Difference Formulas: \(\cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b)\)

Recognizing and utilizing trigonometric identities is essential when simplifying expressions in trigonometry. In our context, they helped in transforming the equation \(r^2 (\cos^2(\theta) - \sin^2(\theta)) = 1\) to \(r^2 \cos(2\theta) = 1\), simplifying the conversion to polar form.

conversion between coordinate systems

Converting between Cartesian and polar coordinates can initially seem tricky, but it's simply a matter of applying the right formulas and understanding each system's logic.

By definition,

• In Cartesian coordinates, a point is described by \(x\) and \(y\).
• In polar coordinates, the same point is described by \(r\) and \(\theta\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

The conversion involves using the relations:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)
• For reverse, \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\)

These equations allow us to switch coordinate systems easily. For example, by substituting the polar expressions for \(x\) and \(y\) into the Cartesian equation \(x^2 - y^2 = 1\), we achieve the equivalent equation in polar form, \(r^2 \cos(2\theta) = 1\).

Understanding these conversions is essential in fields such as physics and engineering, where analyzing patterns from both perspectives can provide deeper insights.