Problem 27 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (4, \pi / 6) $$

Step-by-Step Solution

Answer

Rectangular coordinates: (2√3, 2).

Step 1: Understand the Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured counterclockwise from the positive x-axis. Here, we have \(r = 4\) and \(\theta = \frac{\pi}{6}\).

Step 2: Convert the Angle from Radians to Degrees (Optional)

While not necessary for the conversion itself, converting the angle from radians to degrees can sometimes make visualization easier. The conversion is: \(\frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ\).

Step 3: Use the Conversion Formulas

To convert from polar to rectangular coordinates, use the formulas \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\). For this problem:\(x = 4 \cdot \cos(\frac{\pi}{6})\) and \(y = 4 \cdot \sin(\frac{\pi}{6})\).

Step 4: Calculate the Cosine and Sine

Recall the values: \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) and \(\sin(\frac{\pi}{6}) = \frac{1}{2}\).

Step 5: Compute the Rectangular Coordinates

Substitute the cosine and sine values into the conversion formulas:\(x = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\) and \(y = 4 \cdot \frac{1}{2} = 2\).

Step 6: State the Rectangular Coordinates

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

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent a point in a plane using a distance and an angle. Unlike the more familiar Cartesian coordinates, which denote a point based on its horizontal and vertical distances from the origin, polar coordinates use:

• The radius, denoted as \( r \), which is the direct line distance from the origin to the point.
• The angle, \( \theta \), measured from the positive x-axis, counterclockwise to the line segment that is \( r \).

This system is especially useful in scenarios where rotation and circular motion need to be described.In our example, the point is given in polar form as \((4, \pi/6)\), indicating the point lies on an imaginary circle, 4 units away from the origin, at an angle of \( \pi/6 \) radians from the positive x-axis.

Rectangular Coordinates Conversion

To convert polar coordinates to rectangular coordinates, we use trigonometric relationships. These relationships leverage the fact that the radius \( r \) creates a right triangle with the horizontal and vertical axes.

• Conversion Formulas: The formulas \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \) allow us to calculate the x (horizontal) and y (vertical) components.

For the conversion,

• First, evaluate \( \cos(\theta) \) and \( \sin(\theta) \).
• Multiply \( r \) (the radius) by these trigonometric values to find the Cartesian coordinates \( (x, y) \).

Using our specifics: For \( r = 4 \) and \( \theta = \pi/6 \), which are used to find \( x \) and \( y \) through the above formulas.

Radians and Degrees Conversion

Angles in polar coordinates can be measured in radians or degrees. This conversion can simplify understanding:

• Degrees: The conventional method of measuring angles, familiar in everyday use, where an entire circle equals 360 degrees.
• Radians: A measure used commonly in mathematics, particularly calculus, where 2\( \pi \) radians are equal to a full circle (360 degrees).

To convert, use the formula: Multiply by \( 180/\pi \) to convert from radians to degrees.
In our exercise, \( \pi/6 \) radians converts to \( 30^\circ \), which makes visualizing the angle a bit easier in terms of a circle.

Trigonometric Functions

Trigonometric functions are crucial for these types of coordinate conversions because they relate angles to the ratios of sides of right triangles.

• Cosine (\( \cos \)) : This function provides the ratio of the adjacent side over the hypotenuse for a given angle in a right triangle.
• Sine (\( \sin \)) : It provides the ratio of the opposite side over the hypotenuse.

In our task, we specifically used these values:

• \( \cos(\pi/6) = \frac{\sqrt{3}}{2} \) and \( \sin(\pi/6) = \frac{1}{2} \)
• Using these values translates the point from polar (\(4, \pi/6\)) to rectangular form (\(x, y\)).

These trigonometric functions are often memorized or found in trigonometric tables for their frequent use in coordinate conversion tasks.