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Chapter 9: Problem 44

Convert the rectangular equation to a polar equation. . \(y=5\)

Short Answer

Expert verified
The polar equation is \(r \sin \theta = 5\).

Step by step solution

01

Understanding Rectangular to Polar Conversion

To convert the rectangular equation of the form \(x = a\) or \(y = b\) to a polar equation, we need to understand the relationship between rectangular and polar coordinates. Recall:- \(x = r \cos \theta\)- \(y = r \sin \theta\)
02

Substitute Rectangular Variables with Polar Equivalents

The given equation is \(y = 5\). In polar coordinates, the \(y\)-coordinate can be expressed as \(y = r \sin \theta\). Substitute this into the equation:\[r \sin \theta = 5\]
03

Reformulate the Equation

The equation \(r \sin \theta = 5\) is already in polar form. It represents a horizontal line in rectangular coordinates, which translates into the equation in polar format without additional changes. This is the polar equation corresponding to the original rectangular equation \(y = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Coordinates
Understanding rectangular coordinates is crucial for navigating between different mathematical systems. In rectangular coordinates, any point in a plane is identified by an ordered pair \(x, y\). These coordinates are based on the Cartesian coordinate system, which uses horizontal and vertical lines.
  • The x-coordinate refers to the distance from the y-axis in the horizontal direction.
  • The y-coordinate indicates the distance from the x-axis in the vertical direction.
This system is intuitive as it aligns with our everyday experiences of width and height. It is particularly useful in graphing linear equations like \(y = 5\), where the relationship between variables is straightforward to visualize. Here, it describes a horizontal line that remains constant along the y-axis.
Equation Conversion
Equation conversion often involves translating one form into another to suit different mathematical operations or visualizations. For converting equations from rectangular to polar form, we utilize specific relationships between the coordinates. Specifically:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
The conversion process requires substituting these expressions into the original equation. For example, the equation \(y = 5\) becomes \(r \sin \theta = 5\) when expressed in polar form. This allows for a more comprehensive analysis of the function's behavior in contexts where circular and radial measures make sense.
Trigonometric Substitution
Trigonometric substitution is a method used to simplify the conversion between rectangular and polar equations. By substituting trigonometric identities derived from right triangle relationships, we can reformulate equations to reflect their properties in polar terms.
This involves using identities like:
  • \( \sin \theta\) for the vertical component or y-coordinate
  • \( \cos \theta\) for the horizontal component or x-coordinate
In the given problem, when substituting \(y = r \sin \theta\) into the equation \(y = 5\), we directly achieve \(r \sin \theta = 5\). This straightforward trigonometric substitution allows us to represent the same geometric figure鈥攁 horizontal line鈥攚ithin the polar coordinate system without changing its intrinsic properties.

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Most popular questions from this chapter

Find: (a) \(\frac{d y}{d x}\) (b) the equation of the tangent and normal lines to the curve at the indicated \(\theta\) -value. \(r=1 ; \quad \theta=\pi / 4\)

Numerically approximate the given arc length. Approximate the are length of the parabola \(x=t^{2}-t\) \(y=t^{2}+t\) on [-1,1] using Simpson's Rule and \(n=4 .\)

Parametric equations for a curve are given. Find \(\frac{d^{2} y}{d x^{2}},\) then determine the intervals on which the graph of the curve is concave up/down. \(x=t^{2}-t, \quad y=t^{2}+t\)

Graph the polar function on the given interval. \(r=2+\sin \theta, \quad[0,2 \pi]\)

Create your own polar function, \(r=f(\theta)\) and sketch it. Describe why the graph looks as it does.
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