Converting R = 10 Cos Θ To Rectangular Coordinates A Step-by-Step Guide

In the realm of mathematics, the ability to seamlessly transition between different coordinate systems is a cornerstone of problem-solving. Polar coordinates, with their elegant representation of points using radial distance (r) and angle (θ), offer a unique perspective compared to the familiar rectangular coordinates (x, y). However, many equations initially presented in polar form can be transformed into their rectangular counterparts, unlocking new avenues for analysis and interpretation. This article delves into the process of converting the polar equation r = 10 cos θ into its rectangular form, providing a step-by-step guide and exploring the underlying concepts.

Transforming Polar Equations to Rectangular Coordinates

The transformation from polar to rectangular coordinates hinges on a set of fundamental relationships that connect the two systems. These relationships, derived from basic trigonometry and the Pythagorean theorem, act as the Rosetta Stone for translating between the languages of polar and rectangular coordinates. The key equations are:

• x = r cos θ
• y = r sin θ
• r² = x² + y²

These equations form the bedrock of our transformation process. By strategically employing these relationships, we can systematically eliminate the polar variables (r and θ) and express the equation solely in terms of the rectangular variables (x and y).

Step-by-Step Conversion of r = 10 cos θ

Now, let's embark on the journey of converting the given polar equation, r = 10 cos θ, into its rectangular form. This process involves a series of algebraic manipulations guided by the fundamental relationships mentioned earlier.

Step 1: Multiplying Both Sides by r

The initial step involves multiplying both sides of the equation by 'r'. This seemingly simple maneuver sets the stage for leveraging the crucial relationship r² = x² + y². Multiplying both sides of r = 10 cos θ by 'r', we obtain:

r² = 10r cos θ

This equation now features the term 'r²', which we can directly replace using the Pythagorean relationship.

Step 2: Substituting r² and r cos θ

This is where the magic happens. We now substitute r² with x² + y² and r cos θ with x, based on our fundamental transformation equations. This substitution effectively replaces the polar variables with their rectangular counterparts. The equation transforms into:

x² + y² = 10x

We have successfully eliminated 'r' and 'θ', and the equation is now expressed in terms of 'x' and 'y'. However, to reveal the true nature of the equation, we need to proceed with further algebraic manipulation.

Step 3: Rearranging and Completing the Square

To gain a deeper understanding of the geometric shape represented by the equation, we rearrange the terms and employ the technique of completing the square. This technique allows us to rewrite the equation in a standard form that readily identifies the shape.

First, we move all terms to the left-hand side:

x² - 10x + y² = 0

Now, we focus on the 'x' terms and complete the square. To do this, we take half of the coefficient of the 'x' term (-10), square it ((-5)² = 25), and add it to both sides of the equation. This ensures that the equation remains balanced while allowing us to create a perfect square trinomial:

x² - 10x + 25 + y² = 25

The left-hand side can now be factored into a perfect square:

(x - 5)² + y² = 25

Step 4: Recognizing the Equation of a Circle

The final form of the equation, (x - 5)² + y² = 25, reveals its true identity. This is the standard equation of a circle with a center at (5, 0) and a radius of 5. The equation now provides a clear geometric interpretation of the original polar equation.

The General Form and its Significance

The equation (x - 5)² + y² = 25 represents the rectangular form of the polar equation r = 10 cos θ. This form is particularly insightful because it directly reveals the geometric shape represented by the equation – a circle. The general form of the equation of a circle is:

(x - h)² + (y - k)² = r²

where (h, k) represents the center of the circle and 'r' represents the radius. By comparing our transformed equation with the general form, we can readily identify the center (5, 0) and the radius (5) of the circle.

The transformation from polar to rectangular coordinates not only provides an alternative representation of the equation but also unlocks a deeper understanding of its geometric properties. In this case, the rectangular form clearly reveals the circular nature of the equation, which might not be immediately apparent from its polar form.

Visualizing the Transformation

To further solidify the understanding of this transformation, it's helpful to visualize the relationship between the polar and rectangular representations. The polar equation r = 10 cos θ describes all points where the radial distance 'r' is equal to 10 times the cosine of the angle 'θ'. As 'θ' varies from 0 to π, the equation traces out a circle in the rectangular coordinate plane. The circle is centered at (5, 0) and has a radius of 5, as confirmed by the rectangular form of the equation.

Applications and Extensions

The ability to convert between polar and rectangular coordinates is not merely an academic exercise; it has numerous applications in various fields, including:

• Physics: Describing motion in two dimensions, particularly circular or rotational motion.
• Engineering: Analyzing electrical circuits, signal processing, and control systems.
• Computer Graphics: Creating and manipulating geometric shapes.
• Navigation: Representing positions and directions on maps.

The principles demonstrated in this example can be extended to transform a wide range of polar equations into their rectangular counterparts. The key lies in strategically applying the fundamental relationships and employing algebraic techniques such as completing the square to reveal the underlying geometric shapes.

Conclusion

In this comprehensive guide, we have successfully transformed the polar equation r = 10 cos θ into its rectangular form, (x - 5)² + y² = 25. This process involved a step-by-step application of the fundamental relationships between polar and rectangular coordinates, culminating in the identification of a circle with a center at (5, 0) and a radius of 5. The ability to seamlessly convert between coordinate systems empowers us to analyze equations from different perspectives and unlock a deeper understanding of their geometric properties. This skill is invaluable in various scientific and engineering disciplines, highlighting the importance of mastering coordinate transformations.

By understanding the connection between polar and rectangular coordinates, we gain a more complete understanding of mathematical concepts and their applications in the real world. The transformation of r = 10 cos θ serves as a powerful example of how different mathematical representations can illuminate the same underlying truth, enriching our problem-solving capabilities and broadening our mathematical horizons.