Problem 64 - College Trigonometry

Question

60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \)r\( in all but \)\\# 60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for \)\theta$. $$ x^{2}+y^{2}=25 $$

Step-by-Step Solution

Answer

The polar form of the equation is \(r = \pm 5\).

Step 1: Understanding Polar Coordinates

Polar coordinates are represented by \((r, \theta)\) where \(r\) is the radius or distance from the origin and \(\theta\) is the angle from the positive x-axis. We want to express the given rectangular equation using these coordinates.

Step 2: Relating Rectangular and Polar Coordinates

Recall the conversions: \(x = r \cos(\theta)\), \(y = r \sin(\theta)\), and \(x^2 + y^2 = r^2\). These will help transform the equation \(x^2+y^2=25\) into an equation involving \(r\) and \(\theta\).

Step 3: Apply the Conversion to the Equation

Substitute \(x^2+y^2 = r^2\) into the given equation. Thus, replacing the left side of the equation gives us: \[r^2 = 25\].

Step 4: Solving for \(r\) from \(r^2 = 25\)

Find \(r\) by taking the square root on both sides of the equation: \[r = \pm 5\]. Since \(r\) represents a distance, it can be positive or negative, which indicates different directions along a line through the origin.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system for defining a point in a plane using two values:

• The x-coordinate (abscissa), representing the horizontal distance from the origin.
• The y-coordinate (ordinate), representing the vertical distance from the origin.

These coordinates are determined from two perpendicular axes: the x-axis and the y-axis, creating a grid system.
Rectangular coordinates offer a straightforward method to specify locations, especially when working with geometric shapes and algebraic equations.
In the original exercise, the equation given is in rectangular form: \(x^2 + y^2 = 25\). This is a familiar equation representing a circle centered at the origin with a radius of 5 units.

Coordinate Conversion

To solve many mathematical problems, converting between different coordinate systems is crucial.

• For converting from rectangular to polar coordinates, the key formulas are:
  • x = r \cdot \cos(\theta)
  • y = r \cdot \sin(\theta)
  • x^2 + y^2 = r^2

These equations allow you to switch from x and y coordinates to r and \(\theta\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis.
In our problem, we replace \(x^2 + y^2\) with \(r^2\), resulting in the simplification \(r^2 = 25\), which is much easier to handle in polar coordinates.

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to simplify equations by introducing trigonometric identities.
It's particularly useful when converting between coordinate systems or when dealing with integrals.
In the context of converting rectangular to polar coordinates, trigonometric identities like \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) enable you to substitute and rearrange terms efficiently.
By utilizing these identities, the equation \(x^2 + y^2 = 25\) becomes \(r^2 = 25\). This step is crucial in streamlining the problem-solving process, eliminating complex algebraic manipulation, and directly providing the polar equation.

Polar Equations

Polar equations describe curves using polar coordinates \((r, \theta)\).

• Each point on the plane is determined by:
  • \(r\), the radius or distance from the origin.
  • \(\theta\), the angle measured counterclockwise from the positive x-axis.

In polar form, our original rectangular equation \(x^2 + y^2 = 25\) transforms into \(r^2 = 25\).
Solving for \(r\), we find \(r = \pm 5\).
This indicates that the graph is a circle with a radius of 5 units in the polar plane, centered at the origin.
Polar coordinates are especially useful for circular and spiral shapes, as they naturally accommodate these forms without needing complex expressions.