Question

Convert the rectangular equation to a polar equation. \(y=4 x+7\)

Short answer

The polar equation is \(r = \frac{7}{\sin \theta - 4\cos \theta}\).

Step-by-step solution

Recall the Conversion Formulas

Start by recalling the conversion relationships between rectangular and polar coordinates. The equations are: 1. For transforming from polar to rectangular, use: \(x = r \cos \theta\) and \(y = r \sin \theta\).2. For transforming \((x, y)\) to polar, use: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(\frac{y}{x})\).

Substitute for x and y

Replace \(x\) with \(r \cos \theta\) and \(y\) with \(r \sin \theta\) according to the conversion formulas. This gives:\(r \sin \theta = 4(r \cos \theta) + 7\).

Simplify the Equation

Distribute and rearrange the equation to isolate \(r\). This becomes:\(r \sin \theta = 4r \cos \theta + 7\).

Factor out r

Bring all terms involving \(r\) on one side of the equation:\(r \sin \theta - 4r \cos \theta = 7\).Factor \(r\) out of the terms on the left:\(r(\sin \theta - 4\cos \theta) = 7\).

Solve for r

Isolate \(r\) by dividing each side of the equation by \((\sin \theta - 4\cos \theta)\):\(r = \frac{7}{\sin \theta - 4\cos \theta}\).

Key concepts

Rectangular to Polar Conversion

Converting rectangular coordinates to polar coordinates involves understanding how our positions shift between the two systems. Rectangular coordinates denote a location in the form of

• \((x, y)\)

representing horizontal and vertical distances from a fixed point, usually the origin. In contrast, polar coordinates express a location using

• \((r, \theta)\)

where \(r\) indicates distance from the origin and \(\theta\) is the angle with respect to the positive x-axis.To perform a rectangular to polar conversion, use the formulas:

• \(r = \sqrt{x^2 + y^2}\) for the radial distance
• \(\theta = \tan^{-1}(\frac{y}{x})\) for the angle

When dealing with equations, we substitute \(x\) and \(y\) into their respective polar forms. This often results in simpler relationships more natural to circular motion, providing a different perspective on the same physical phenomena.

Rectangular Coordinates

Rectangular coordinates provide a systematic way to represent positions on a plane using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this system is identified by its

• horizontal distance from the y-axis
• vertical distance from the x-axis

This coordinate system is also known as the Cartesian coordinate system, named after René Descartes, who formalized its use. It's especially useful in defining linear graphs or plotting equations in two dimensions, such as

• \(y = 4x + 7\)

In this equation, \(x\) represents the independent variable moving along the horizontal line, and \(y\) changes correspondingly along the vertical line. This concept relies heavily on the understanding of thinking in right angles, allowing both theoretical and practical explorations in mathematics and physics.

Trigonometric Functions

Trigonometric functions are the core tools for connecting geometry to algebra through angles and calculations involving circular measures. These functions relate an angle to ratios of sides in a right-angled triangle. The most common trigonometric functions are:

• Sine \((\sin \theta)\)
• Cosine \((\cos \theta)\)
• Tangent \((\tan \theta) = \frac{\sin \theta}{\cos \theta}\)

In transforming rectangular to polar equations, sine and cosine facilitate the replacement of \(x\) and \(y\) components:

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

These transformations help express relationships in simpler polar forms, crucial in fields like engineering and physics where angles and rotational motions often play a key role. Understanding these functions in the context of the unit circle can also deepen insights into periodic phenomena and harmonic motions.