Problem 48 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar form is \(r^2 = \sec(2\theta)\).

Step 1: Recall the Cartesian to Polar Coordinates Conversion Formulas

Polar coordinates \((r, \theta)\) relate to Cartesian coordinates \((x, y)\) through the equations: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). We'll use these to convert the given Cartesian equation to polar form.

Step 2: Substitute the Polar Formulas into the Equation

Replace \(x\) with \(r \cos(\theta)\) and \(y\) with \(r \sin(\theta)\) in the equation \(x^2 - y^2 = 1\). This gives:\((r \cos(\theta))^2 - (r \sin(\theta))^2 = 1\).

Step 3: Apply Algebraic Simplification

Simplify the equation: \[r^2 (\cos^2(\theta) - \sin^2(\theta)) = 1\]\(\cos^2(\theta) - \sin^2(\theta)\) is a trigonometric identity equal to \(\cos(2\theta)\).

Step 4: Finalize the Equation in Polar Form

Substitute \(\cos(2\theta)\) into the simplified equation from Step 3:\[r^2 \cos(2\theta) = 1\]Solve for \(r^2\) to get the polar form of the equation:\[r^2 = \frac{1}{\cos(2\theta)}\], or equivalently, \[r^2 = \sec(2\theta)\].

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a system of assigning ordered pairs

• The system identifies the position of a point in a plane.
• Each point is represented by two values: \(x\), a horizontal displacement, and \(y\), a vertical displacement.

Imagine a graph sheet you might use in math class.
The two axes, x and y, intersect to form a grid.
Each point on this grid expresses exactly where something is.
In the math problem we have, the equation \(x^{2} - y^{2} = 1\) is described using these coordinates.To solve, we need to change this format into polar coordinates, where a point is defined as \( (r, \theta)\).
This allows us to describe the same point in terms of distance and angle.Polar coordinates can offer a clearer understanding in problems involving circular or rotational movement.

Trigonometric Identities

Trigonometric identities are crucial in simplifying equations.

• These identities provide relationships between trigonometric functions such as sine, cosine, and tangent.
• They allow for the simplification or transformation of expressions.
• A key identity is \(\cos^{2}(\theta) - \sin^{2}(\theta) = \cos(2\theta)\), which is used to simplify the polar form exercise.

Recognizing and applying identities can make complex mathematical expressions much easier to manage.
For example, after substituting polar coordinate expressions, we applied \(\cos^{2}(\theta) - \sin^{2}(\theta)\) to rewrite the equation in a simpler form.
In many cases, these identities allow us to uncovered deeper relationships within mathematical expressions.
Remember, mastering these trigonometric identities can not only enhance problem-solving skills but also deepen your understanding of trigonometry itself.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form.

• Simplification can involve combining like terms, factoring, or applying specific identities.
• Through simplification, complex problems become easier and clearer.

In our exercise, once the equation was re-written using polar coordinates, it required simplification.
Remember that every algebraic step should be logical - following rules of operations and using identities where applicable.
For example, after using the identity \(\cos(2\theta)\), the equation simplified to \(r^2\cos(2\theta) = 1\), which eventually leads to \(r^2 = \sec(2\theta)\).This clear form is immensely helpful as it expresses the given equation efficiently.
Understanding simplification helps not only in current subject material but will also provide a solid foundation for solving more complex equations in advanced mathematics.