Problem 25 - Algebra and Trigonometry

Question

A point is graphed in polar form. Find its rectangular coordinates. $$ (4, \pi / 6) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((2\sqrt{3}, 2)\).

Step 1: Understand Polar Coordinates

Polar coordinates represent a point in the plane using the distance from the origin and an angle from the positive x-axis. The given polar coordinates are \((r, \theta) = (4, \pi/6)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle in radians.

Step 2: Use Conversion Formulas

To find the rectangular coordinates \((x, y)\), use the formulas: \[ x = r \cos \theta \] and \[ y = r \sin \theta \]. These formulas convert polar coordinates to Cartesian coordinates.

Step 3: Calculate the x-coordinate

Substitute the values of \(r\) and \(\theta\) into the formula for \(x\): \[ x = 4 \cos(\pi / 6) \]. Using the known value \(\cos(\pi / 6) = \sqrt{3}/2\), calculate \[ x = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \].

Step 4: Calculate the y-coordinate

Now substitute the values of \(r\) and \(\theta\) into the formula for \(y\): \[ y = 4 \sin(\pi / 6) \]. Using the known value \(\sin(\pi / 6) = 1/2\), calculate \[ y = 4 \times \frac{1}{2} = 2 \].

Step 5: Write the Rectangular Coordinates

Combine the values of \(x\) and \(y\) to express the rectangular coordinates: \((x, y) = (2\sqrt{3}, 2)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points on a two-dimensional plane using a mix of distance and direction. Instead of using the familiar x and y axes, they rely on two values:

• Radius (\( r \)) - This is the distance from the origin (center of the coordinate plane) to the point.
• Angle (\( \theta \)) - This measures the direction of the point relative to the positive x-axis, usually in radians.

Imagine a line stretching from the center outward. The radius tells you how far to go along this line, while the angle shows you which way to point the line. In our example, the polar coordinate \((r, \theta) = (4, \pi/6)\) means the point is 4 units from the origin, in the direction \(\pi/6\) radians from the positive x-axis.

Polar coordinates are particularly useful in scenarios where problems involve rotations or circular motion. They might seem complex at first glance, but they often simplify calculations related to angles and distances on a curved path.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the conventional system most people are familiar with. They specify a point's position using two perpendicular axes:

• x-coordinate - Measures how far left or right a point is from the y-axis.
• y-coordinate - Measures how far up or down a point is from the x-axis.

Each point is described as \((x, y)\), where 'x' shows the horizontal placement, and 'y' shows the vertical positioning. This system is commonly used because it aligns well with how we view and draw graphs on paper or computer screens.

In science and engineering, rectangular coordinates simplify tasks like plotting lines, calculating slopes, or finding distances between points. They map directly onto the grid system, making them intuitive for linear relationships and right-angle calculations. Converting from polar to rectangular coordinates can make working in these contexts easier, illustrating where points lie in an intuitive, grid-style format.

Trigonometric Functions (Sine and Cosine)

Trigonometric functions like sine and cosine play a crucial role in converting polar coordinates to rectangular coordinates. These functions effectively relate the angles and sides of triangles, forming connections between circular geometry and linear dimensions.

The basic formulas are:

• Cosine (\(\cos \theta\)) - Relates the adjacent side of a triangle to its hypotenuse, giving the x-coordinate when combined with the radius in polar coordinates:\[ x = r \cos \theta \]
• Sine (\(\sin \theta\)) - Relates the opposite side of a triangle to its hypotenuse, providing the y-coordinate using the formula:\[ y = r \sin \theta \]

Trigonometry taps into the underlying geometry of a point’s position and can be essential in practical applications such as navigation, signal processing, or even the computation of Fourier transforms. In our example, we used \(\cos(\pi/6) = \sqrt{3}/2\) and \(\sin(\pi/6) = 1/2\) to find our rectangular coordinates as \((2\sqrt{3}, 2)\). These calculations provide a bridge from the abstract circular motion view of polar coordinates to the more familiar, tangible format of a standard cartesian plane.