Question

Convert the polar equation to a rectangular equation. \(\theta=\pi / 6\)

Short answer

The rectangular equation is \(y = \frac{\sqrt{3}}{3}x\).

Step-by-step solution

Understanding the Polar Equation

The given polar equation is \(\theta = \frac{\pi}{6}\). Recall that in polar coordinates, \(\theta\) represents the angle counterclockwise from the positive x-axis.

Conversion Formula

To convert from polar to rectangular coordinates, we use the conversion formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta)\). We also use the relation \(\tan(\theta) = \frac{y}{x}\).

Applying the Tangent Formula

Given \(\theta = \frac{\pi}{6}\), we use the identity \(\tan(\theta) = \frac{y}{x}\). Since \(\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\), we write \(\frac{y}{x} = \frac{\sqrt{3}}{3}\).

Forming the Rectangular Equation

Multiplying both sides of \(\frac{y}{x} = \frac{\sqrt{3}}{3}\) by \(x\) gives \(y = \frac{\sqrt{3}}{3}x\). This equation is in the form \(y = mx\) where \(m\) is the slope.

Key concepts

Polar Coordinates

Polar coordinates are a unique way to locate points based on their distance and direction from a central point, known as the origin. In polar coordinates, each point is described using a pair

• the distance from the origin, represented as \( r \)
• the angle \( \theta \) from the positive x-axis.

The angle is measured in the counterclockwise direction. This system is handy in scenarios involving circular motion, spirals, or any situation where direction and distance are crucial. Polar coordinates simplify the representation of curves like spirals and circles, making them immensely useful in physics and engineering. When a given problem uses polar coordinates, remember that \( \theta = \frac{\pi}{6} \) is equivalent to a 30-degree angle. This is often a standard reference in many trigonometric problems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are another method of identifying the position of a point in a plane. Instead of using distance and angle, this system uses two numerical values

• \( x \) for the horizontal distance
• \( y \) for the vertical distance

These coordinates form a grid or plane. It is the most commonly used coordinate system in mathematics, allowing easy graphing of linear equations and plotting points based on their horizontal and vertical distances from the origin. Rectangular coordinates are particularly useful when calculating and visualizing slopes and intercepts of lines, as it involves straightforward arithmetic operations. They are the backbone for algebraic graphing techniques in both 2D and 3D spaces.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions like sine, cosine, and tangent, which hold true for every value of the variables where both sides are defined. They form relationships that provide handy ways to convert between angles and ratios. Some basic identities crucial in coordinate conversions include:

• \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)
• \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
• \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \text{ or } \frac{\sqrt{3}}{3} \)

These identities are invaluable in bridging polar and rectangular coordinates because they allow converting angles to their geometric component ratios. In our step-by-step solution, \( \tan(\theta) = \frac{y}{x} \) was used directly to link the polar angle to the rectangular equation form.

Slope of a Line

The slope of a line is an important concept in Cartesian geometry. It describes the steepness and direction of a line and is denoted by \( m \) in the equation of a line, which in slope-intercept form is \( y = mx + b \). The slope is calculated as the change in the vertical direction (rise) divided by the change in horizontal direction (run), expressed as \( m = \frac{\Delta y}{\Delta x} = \frac{y}{x} \) for a straight line passing through the origin.

In the given problem, when converting the tangent of an angle to a rectangular form, we effectively determined the slope of the line. Here, \( \tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3} \) provided the slope \( m \). This means that for every unit increase in \( x \), \( y \) increases by \( \frac{\sqrt{3}}{3} \), resulting in the line equation \( y = \frac{\sqrt{3}}{3}x \). This slope-intercept form makes it easy to graph such lines and understand their behavior in 2D space and highlights the line's inclination compared to axes.