Problem 56 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$r=4$$

Step-by-Step Solution

Answer

The rectangular form of the polar equation \(r = 4\) is \(x^2 + y^2 = 16\).

Step 1: Identify the given polar equation

The given polar equation is \(r = 4\).

Step 2: Use Pythagorean theorem to convert into rectangular form

In rectangular coordinates, any point can also be represented using the Pythagorean theorem as \(x^2 + y^2 = r^2\). Substitute \(r = 4\) to get the equation in rectangular form.

Step 3: Write the final equation

Substituting \(r = 4\) into the formula \(x^2 + y^2 = r^2\), we get \(x^2 + y^2= 16\). This is the rectangular form of the given polar equation \(r = 4\).

Key Concepts

Polar Coordinates

In mathematics, polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point, known as the pole, and an angle from a reference direction. The pole corresponds to the origin in the rectangular coordinate system, and the reference direction is usually the positive x-axis. A point in polar coordinates is represented as \( (r, \theta) \), where \( r \) is the radius or the distance from the pole, and \( \theta \) is the angle measured in radians from the positive x-axis.

To visualize this, imagine standing at the center of a circle and pointing to a spot on the circle's edge. The distance at which you reach out is \( r \) and the angle your arm makes with a fixed line, such as due east, is \( \theta \) in polar coordinates. This system is particularly useful for dealing with problems involving circular or spiral patterns, where the distances from a central point are more relevant than distances along perpendicular axes, as in rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular coordinate axes, traditionally labeled as the x and y axes, to define the position of a point in a plane. The position is given by an ordered pair \( (x, y) \), indicating the point’s horizontal and vertical distances from the origin, which is where the x and y axes intersect.

Every point on the plane can be associated with a unique pair of numbers in rectangular coordinates, making it a straightforward and intuitive system to represent geometric shapes and algebraic equations. Consider a grid where you count steps right or left for \( x \) and steps up or down for \( y \) to get to a point. Rectangular coordinates are versatile and widely used in geometry, calculus, and various fields of engineering and physics.

Pythagorean Theorem

The Pythagorean theorem is a foundational principle in Euclidean geometry, named after the ancient Greek mathematician Pythagoras. It relates the lengths of the sides of a right-angled triangle. Specifically, the theorem states that the square of the length of the hypotenuse \( (c) \) is equal to the sum of the squares of the lengths of the other two sides \( (a) \) and \( (b) \), or in mathematical terms, \[ a^2 + b^2 = c^2 \].

For our discussion on coordinate conversion, the Pythagorean theorem is used to connect polar and rectangular coordinates. Considering a point in the plane, \( r \) represents the hypotenuse of a right triangle, with the point’s \( x \) and \( y \) coordinates representing the other two sides. Thus, if you know the radius and angle of a point in polar coordinates, you can find its position in rectangular coordinates using \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \), thereby applying the concepts behind the Pythagorean theorem.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped right circular cone. Depending on the angle of the plane's intersection with the cone, the resulting curve can be a circle, ellipse, parabola, or hyperbola. Each of these has a specific equation and properties.

The circle, for example, is an important conic section which is represented in polar form as \( r = \text{constant} \) and in rectangular coordinates as the equation \( x^2 + y^2 = r^2 \), where \( r \) is the radius. This form of the circle’s equation comes directly from the Pythagorean theorem, asserting the uniform distance from the origin to every point on the circle's perimeter, which remains the same, hence the constant radius. When converting from polar to rectangular coordinates, recognizing the nature of these curves can greatly simplify the process and allow for a deeper understanding of the geometric relationships involved.