Problem 73 - Precalculus

Question

Convert the rectangular equation to polar form. Assume \(a > 0\). $$y=x$$

Step-by-Step Solution

Answer

The polar form of the given rectangular equation \(y=x\) is \(tan(\theta) = 1\).

Step 1: Understand Rectangular and Polar Coordinates Relationships

The polar coordinates \((r,\theta)\) and rectangular coordinates \((x, y)\) can be correlated using following two equations, \(r = \sqrt{x^2 + y^2}\) and \(\theta = arctan(\frac{y}{x})\) when x>0. Based on this, we'll convert given rectangular equation \(y=x\) into polar form.

Step 2: Substitute Y

We can substitute \(y\) by \(r*sin(\theta)\) in the given equation \(y=x\) because \(y= r*sin\theta\), it's an identity that relates polar and rectangular coordinates. So after the substitution we get \(r*sin(\theta) = x\).

Step 3: Substitute X

Now substitute \(x\) by \(r*cos(\theta)\) in equation \(r*sin(\theta) = x\) since \(x= r*cos\theta\), it's an identity that relates polar and rectangular coordinates. So after substitution we get \(r*sin(\theta) = r*cos(\theta)\).

Step 4: Simplify Equation

We can simplify the equation by dividing every term by \(r*cos(\theta)\). This will leave us with the polar form of the equation: \(tan(\theta) = 1\). This is the final answer which represents the rectangular equation \(y=x\) in polar coordinates.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are familiar to anyone who has worked on a standard x-y plane. In this system, a point is represented by two numbers: its horizontal placement, denoted as \(x\), and its vertical placement, denoted as \(y\). For instance, the point \((3, 4)\) means the point is located 3 units along the positive x-axis and 4 units along the positive y-axis. The geometric interpretations relying on these coordinates are straightforward and enable us to plot linear equations, like \(y = x\), as straight lines. Here, each point on the line has equal x and y values, visualizing a 45-degree incline linearly through the origin (0,0).Since rectangular coordinates are rooted in right-angle projections, they provide an intuitive framework for aligning and measuring distances in geometrical and algebraic contexts. They are immensely practical for analyses involving direct real-world dimensions.

Equation Conversion

Equation conversion is the process of altering the representation of equations from one coordinate system to another. For instance, transforming the equation \(y = x\) from rectangular form into polar form involves identifying corresponding points in each system.To convert equations, we employ relationships between the coordinate systems. Notably for polar coordinates, \(x\) and \(y\) are expressed in terms of \(r\) and \(\theta\), where \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). By substituting these into the rectangular equation, the expression aligns with polar conventions. Through our example \(y = x\), substituting \(x\) and \(y\) with their polar equivalents and simplifying the resulting equation yields a different perspective on the mathematical concept. This aids in a broader understanding of how the same geometrical line can be conceptualized in varying formats.

Trigonometry Identities

Trigonometry identities form a vital backbone in converting equations between coordinate systems as they precisely relate angles to side lengths in a circle. Essential identities include:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

For our conversion, the identity \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) was pivotal. By expressing both \(x\) and \(y\) in terms of sin and cos, we form a bridge across which equations travel between rectangular and polar forms seamlessly.In this context, recognizing that \(\tan(\theta) = 1\) implies that the points along the angle bisector of quadrants where x equals y, deepens our understanding of the angles at which relationships hold true. It shows how trigonometric reasoning supplements geometric analysis.

Coordinate Systems

Coordinate systems are frameworks for identifying and magnifying relationships within the space. Rectangular and polar coordinates offer different tools, analogous to vantage points, for describing positions.While rectangular coordinates are prevalent for direct navigation based on perpendicular axis alignments, polar coordinates part information radial from a central point \((r, \theta)\). Why use different systems?

• Rectangular systems suit structures and grids.
• Polar systems suit rotations and spirals, common in scenarios such as navigation or physics regarding forces acting through angles and distances.

Understanding different coordinate systems fosters flexibility and lends to enhanced analytical capabilities when tackling problems, including those where properties like symmetry or angles, over-position, are critical.