Problem 79 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ r=\sqrt{2} $$

Step-by-Step Solution

Answer

The equation in rectangular coordinates is \( x^2 + y^2 = 2 \).

Step 1: Understand Polar Coordinates

In polar coordinates, the position of a point is determined by the distance from the origin, denoted as \( r \), and the angle \( \theta \) measured from the positive x-axis. The given equation \( r = \sqrt{2} \) specifies a circle centered at the origin with radius \( \sqrt{2} \).

Step 2: Use Polar to Rectangular Coordinate Conversion Formulas

The relations that convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\) are: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Here, we are given that \( r = \sqrt{2} \).

Step 3: Substitute r in Rectangular Formulas

Substitute \( r = \sqrt{2} \) into the conversion formulas: \( x = \sqrt{2} \cos(\theta) \) and \( y = \sqrt{2} \sin(\theta) \).

Step 4: Derive the Rectangular Coordinates Equation

Recall the Pythagorean identity: \( \cos^2(\theta) + \sin^2(\theta) = 1 \). By squaring both equations from Step 3, we have: \( x^2 = 2 \cos^2(\theta) \) and \( y^2 = 2 \sin^2(\theta) \). Add these equations: \( x^2 + y^2 = 2(\cos^2(\theta) + \sin^2(\theta)) \).

Step 5: Simplify and Obtain Final Equation

Using the identity from the previous step, we substitute \( \cos^2(\theta) + \sin^2(\theta) = 1 \), leading to \( x^2 + y^2 = 2 \). This is the equation of the circle in rectangular coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing a point in a plane based on its distance from a reference point and its angle from a reference direction. Unlike rectangular coordinates, which rely on horizontal and vertical distances (or x and y values), polar coordinates use:

• r: the radius or distance of the point from the origin.
• θ: the angle from the positive x-axis to the line joining the origin with the point.

For example, the polar equation given in our problem is \( r = \sqrt{2} \). This indicates that every point maintaining a constant distance of \( \sqrt{2} \) from the origin forms a circle. Each point on this circle can be identified by varying \( \theta \) but always keeping \( r \) constant. This representation can be highly useful in cases like waves and rotations, where the relationship with the center and angle is more intuitive than using straight lines as reference.

Coordinate Conversion

Coordinate conversion involves translating points from polar to rectangular coordinates and vice versa, to suit different problem-solving scenarios. To move from polar to rectangular coordinates, we use the following conversion formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

Hence, if given \( r = \sqrt{2} \) and utilizing these transformation formulas, we find:

• \( x = \sqrt{2} \cos(\theta) \)
• \( y = \sqrt{2} \sin(\theta) \)

In this scenario, by squaring both expressions, we prepare for further simplification into rectangular coordinates. This process is about reframing the same spatial information into another format, helping visual or computational understanding depending on the problem's needs.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric equation that establishes a constant relationship between sine and cosine.It states that \( \cos^2(\theta) + \sin^2(\theta) = 1 \). This identity is central in translating from polar to rectangular coordinates.
Using it simplifies the rectangular equation derived from polar terms. When working with polar conversion, after substituting for \( x \) and \( y \) using polar coordinates, we encounter:

• \( x^2 = 2 \cos^2(\theta) \)
• \( y^2 = 2 \sin^2(\theta) \)

By adding these squares together, \( x^2 + y^2 = 2(\cos^2(\theta) + \sin^2(\theta)) \), and substituting the Pythagorean identity, we simplify the equation to \( x^2 + y^2 = 2 \).
Ultimately, understanding and applying the Pythagorean identity results in efficiently achieving the rectangular coordinates equation. It highlights the circular relationship inherent in the function, grounding it in familiar geometry.