Problem 25 - Precalculus with Limits

Question

In Exercises 19-28, a point in polar coordinates is given. Convert the point to rectangular coordinates. \(\left(-2, 7\pi/6\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates of the point \(-2, 7\pi/6\) in polar coordinates are \(\sqrt{3}, 1\).

Step 1: Identify the values of r and θ

From the given polar coordinate (-2, 7π/6), the value of r is -2 and the value of θ is 7π/6.

Step 2: Calculate x coordinate

Use the formula to calculate x: \(x = r * cos(θ)\). Substitute r and θ into the formula: \(x = -2 * cos(7π/6) = -2 * (-sqrt{3}/2) = sqrt{3}\). So, x = sqrt{3}.

Step 3: Calculate y coordinate

Use the formula to calculate y: \(y = r * sin(θ)\). Substitute r and θ into the formula: \(y = -2 * sin(7π/6) = -2 * (-1/2) = 1\). So, y = 1.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of using a grid with x and y coordinates like in rectangular coordinates, polar coordinates use:

• an angle \(\theta\), measured from the positive x-axis, to specify direction
• and a radius \(r\), which tells you how far away from the origin the point is.

For example, in the given exercise, the polar coordinate \((-2, \frac{7\pi}{6})\) means that the point is at a distance of 2 units from the origin, and it is located at an angle of \(\frac{7\pi}{6}\) radians. Note that the negative radius reverses the direction, effectively pointing the opposite direction to the initial angle.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use a grid-like system to specify the location of points. This system includes two perpendicular lines known as:

• x-axis: the horizontal line
• y-axis: the vertical line

Each point is defined by a pair of values, \(x\) and \(y\), which indicate how far the point is from the origin along each axis. With the conversion from polar to rectangular coordinates, we aim to find these \(x\) and \(y\) values that correspond to the same point in the plane. In the case of the exercise, the resulting rectangular coordinates come out to be \((\sqrt{3}, 1)\).

Trigonometric Functions

Trigonometric functions like cosine and sine are crucial in converting between polar and rectangular coordinates. They help to determine:

• the horizontal distance \(x\) (using cosine), and
• the vertical distance \(y\) (using sine)

from the origin, based on the angle \(\theta\). In our example, we used the formulas:

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

By substituting values from our polar input, \(x\) was calculated to be \(\sqrt{3}\) and \(y\) was calculated to be 1, effectively plotting the equivalent point in the rectangular coordinate system.

Coordinate Transformation

Coordinate transformation is all about changing the system used to describe a point while maintaining its actual position. When transforming from polar to rectangular coordinates:

• Use \(x = r \cdot \cos(\theta)\) to find the horizontal distance.
• Use \(y = r \cdot \sin(\theta)\) to find the vertical distance.

The exercise's solution provides a clear example of this process. By knowing the value of \(r\) and \(\theta\), you can apply the trigonometric formulas to get the equivalent rectangular coordinates. This transformation is vital in situations where different perspectives or systems are more useful, such as in physics, engineering and navigation.