Problem 22 - Precalculus with Calculus Previews

Question

Find the rectangular coordinates for each point with the given polar coordinates. $$ (10,-4 \pi / 3) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((-5, 5\sqrt{3})\).

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \((r, \theta)\) where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. In this exercise, the given polar coordinates are \((10, -4\pi/3)\), meaning the point is at a distance of 10 units from the origin and at an angle of \(-4\pi/3\) radians.

Step 2: Convert Polar Angle to Standard Range

Angles in the polar coordinate system can be negative, representing rotation in the clockwise direction. To simplify calculations, it's often easier to convert this to a positive angle\(-4\pi/3\) is equivalent to adding \(2\pi\) to find an equivalent positive angle: \[ \theta = -4\pi/3 + 2\pi = 2\pi/3 \]

Step 3: Use Conversion Formulas

To convert from polar coordinates to rectangular coordinates (x, y), we use the formulas: \[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]where \(r = 10\) and now \(\theta = 2\pi/3\).

Step 4: Calculate the X-Coordinate

Compute the x-coordinate using the formula \(x = r \cos(\theta)\): \[ x = 10 \cos(2\pi/3) \]Now use the known value that \(\cos(2\pi/3) = -1/2\): \[ x = 10 \times (-1/2) = -5 \]

Step 5: Calculate the Y-Coordinate

Compute the y-coordinate using the formula \(y = r \sin(\theta)\): \[ y = 10 \sin(2\pi/3) \]Now use the known value that \(\sin(2\pi/3) = \sqrt{3}/2\): \[ y = 10 \times (\sqrt{3}/2) = 5\sqrt{3} \]

Step 6: Write the Rectangular Coordinates

The rectangular coordinates obtained from the polar coordinates \((10, -4\pi/3)\) are \((-5, 5\sqrt{3})\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a distance and an angle. They are written as \((r, \theta)\) where:

• \(r\) is the radial distance from a point to the origin, similar to the radius of a circle.
• \(\theta\) is the angle measured in radians from the positive x-axis, going counterclockwise.

This system is particularly useful in situations where relationships are easier to understand in terms of angles and distances rather than simple x and y coordinates as in rectangular coordinates.
For example, the polar coordinate \((10, -4\pi/3)\) tells us that the point lies 10 units away from the origin. The angle \(-4\pi/3\) signifies the point's direction is rotated clockwise from the positive x-axis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in the plane using an x-coordinate and a y-coordinate. This system forms a grid defined by two perpendicular axes (x-axis and y-axis). Each point on this grid is determined by:

• x-coordinate: The horizontal distance from the y-axis.
• y-coordinate: The vertical distance from the x-axis.

These coordinates offer a straightforward way to represent and locate points, focusing on horizontal and vertical movement. It's a fundamental part of many branches of mathematics and engineering.

Trigonometric Functions

Trigonometric functions are mathematical functions based on angles that relate the angles of a triangle to the lengths of its sides. In physics and engineering, they're essential for calculations involving waves, oscillations, and circular motion. The primary trigonometric functions are:

• Sine function \(\sin(\theta)\): Ratio of the length of the opposite side to the hypotenuse.
• Cosine function \(\cos(\theta)\): Ratio of the length of the adjacent side to the hypotenuse.
• Tangent function \(\tan(\theta)\): Ratio of the sine to the cosine of an angle \((\sin(\theta)/\cos(\theta))\).

These functions can convert polar coordinates into rectangular coordinates by projecting the radial distance \(r\) onto the x and y axes using the specified angle \(\theta\).

Angle Conversion

Angle conversion is the process of changing an angle's value from one measurement to another. In the context of converting polar coordinates to rectangular coordinates, we frequently adjust negative angles to positive ones to simplify calculations. This is done by:

• Adding or subtracting full circle rotations \((2\pi\) radians or 360 degrees) as needed.

For example, the angle \(-4\pi/3\) can be converted to \(2\pi/3\) by adding \(2\pi\). This allows us to utilize well-known trigonometric values for common angles, streamlining the conversion process.

Coordinate Systems

Coordinate systems are frameworks used to define the position of a point in space. Two of the most common are the polar and rectangular coordinate systems. Each has its advantages depending on the application:

• Polar Coordinate System: Ideal for circular and rotational problems, where the relationship is clearer using angles and distances.
• Rectangular Coordinate System: Best for linear mapping and when precision in direction and distance along straight lines is needed.

Choosing the right coordinate system is crucial for simplifying mathematical modeling and problem-solving. Understanding their conversions helps in switching between perspectives smoothly.