Problem 67 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$\sec \theta=2$$

Step-by-Step Solution

Answer

The rectangular equation is \( y = x \sqrt{3} \).

Step 1: Convert Secant to Cosine

The equation given is in terms of secant: \( \sec \theta = 2 \). Start by converting secant to cosine, because secant is the reciprocal of cosine. Thus, \( \sqrt{1 - (\sin^2 \theta)} = 2 \), which simplifies to \( \cos \theta = \frac{1}{2} \).

Step 2: Find Corresponding x and y Coordinates

In polar coordinates, we have the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). Since \( \cos \theta = \frac{1}{2} \), multiply both sides by \( r \) to get \( x = \frac{r}{2} \).

Step 3: Express r in terms of x

We have the expression for \( x \) from the previous step: \( x = \frac{r}{2} \), which can be rearranged to express \( r \) in terms of \( x \). So, \( r = 2x \).

Step 4: Substitute r into Polar Equation for y

Since \( r = 2x \), we substitute this back into the relationship for \( y \), which is \( y = r \sin \theta \). We know that \( \sin^2 \theta = 1 - \cos^2 \theta \), hence \( \sin \theta = \sqrt{1 - \left( \frac{1}{2} \right)^2} = \frac{\sqrt{3}}{2} \). Thus, \( y = 2x \cdot \frac{\sqrt{3}}{2} = x \sqrt{3} \).

Step 5: Equation in Rectangular Coordinates

Finally, substitute back to finish converting to rectangular coordinates: \( x = \frac{r}{2} \), so the equation already accounts for this relationship in the rectangular system. Hence, the polar equation \( \sec \theta = 2 \) converts to the rectangular coordinates equation: \( y = x \sqrt{3} \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance from a fixed point and an angle from a fixed direction. This fixed point is called the pole, usually located at the origin, and the fixed direction is the positive x-axis. In the polar coordinate system, a point \((r, \theta)\) is located by moving a distance \(r\) from the pole along a direction formed by the angle \(\theta\).

The angle \(\theta\) is typically measured in radians or degrees, and the distance \(r\) is always a non-negative value. This method is especially beneficial when dealing with problems involving circles or waves as circles have a natural representation in polar form. For instance, a circle with radius 1 centered at the origin is simply \(r = 1\) in polar coordinates.

Polar coordinates are especially popular in fields like physics and engineering where systems are rotational or circular by nature.

Rectangular Coordinates

Rectangular coordinates, more commonly known as Cartesian coordinates, represent points on a plane using two numbers: \((x, y)\). Here, \(x\) represents the horizontal displacement, while \(y\) indicates the vertical displacement from the origin. This system is extremely useful for describing positions and motions in a straightforward, linear manner.

For any given point, its position is computed relative to two perpendicular axes intersecting at the origin. This system is beneficial because of its simplicity and its ability to easily translate complex shapes and functions into graphs or mathematical functions.

In the exercise, transforming the coordinates from polar to rectangular involves identifying the right \(x\) and \(y\) values using the angle \(\theta\) and the distance \(r\).

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and sides of triangles. They're critical tools in solving problems involving angles and they facilitate conversions between different coordinate systems.

Some key trigonometric identities include the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). This identity was used in the step-by-step solution, specifically when calculating \(\sin \theta\) from \(\cos \theta\). Other important identities are the reciprocal identities, such as \(\sec \theta = \frac{1}{\cos \theta}\), which directly relate to the original equation \(\sec \theta = 2\).

Understanding these identities helps in seamlessly transitioning between polar and rectangular coordinates, as they provide the necessary trigonometric transformations.

Coordinate Systems

Coordinate systems are frameworks used to uniquely determine the position of a point or other geometric element in a space. Each system has its own unique advantages and is suited to particular types of problems.

• Polar Coordinate System: Ideal for problems involving circular or rotational symmetry, as it simplifies the expression of circular paths.
• Rectangular (Cartesian) Coordinate System: Preferred for linear problems and scenarios requiring simple, unambiguous representations of locations.

Polar and rectangular coordinate systems serve different purposes. They are often interchangeable through mathematical transformations, such as converting an angle \(\theta\) and a distance \(r\) in polar coordinates to \(x\) and \(y\) in rectangular coordinates via formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\).

By understanding both systems, one can efficiently switch between them to leverage their respective advantages depending on the context of the problem being solved.