Problem 27 - Precalculus

Question

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(-2,7 \pi / 6)$$

Step-by-Step Solution

Answer

The rectangular coordinates for the given polar coordinates \(-2, \frac{7\pi}{6}\) are \( \sqrt{3}, -1\).

Step 1: Applying cos formula

Apply the formula to find the x-coordinate of the point in rectangular coordinates. Remember that \(x = r \cos \theta\). Substitute the given values of r and \(\theta\) to get \(x = -2 \cos \left(\frac{7\pi}{6}\right)\). When calculated, this will provide the x-coordinate.

Step 2: Applying sin formula

Repeat this process using the sin function to find the y-coordinate. \(y = r \sin \theta\). So, \(y = -2 \sin \left(\frac{7\pi}{6}\right)\). Upon solving, you will get the y-coordinate.

Step 3: Final Answer

The calculated values of x and y in steps 1 and 2 give the rectangular coordinates of the given point in polar coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using two values: the radial distance and the angular direction. Imagine standing at the center of a circle. The first value, often denoted as \(r\), tells you how far you need to walk in a straight line from the center. The second value, \(\theta\), is an angle that tells you which direction to head in walking this distance.

• \(r\): Radial distance from the origin (center point).
• \(\theta\): Angle in radians, often measured from the positive x-axis.

Using polar coordinates is very helpful in situations where directions and distances are more intuitive, like when dealing with circular motion. The exercise you see involves converting these to more commonly used rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a more familiar way to locate points in a plane using two perpendicular axis: x and y. By specifying how far to the right or left a point is from a vertical line (x-coordinate) and how far up or down it is from a horizontal line (y-coordinate), any point can be described efficiently.

• \(x\): The horizontal distance from the origin along the x-axis.
• \(y\): The vertical distance from the origin along the y-axis.

In our exercise, we start with a point \((-2, \frac{7\pi}{6})\) in polar coordinates, and our goal is to express this point as \((x, y)\). Transforming polar coordinates to rectangular coordinates involves using trigonometric functions since these functions bridge the gap between the angles and distances in both systems.

Trigonometric Functions

Trigonometric functions provide the mathematical tools necessary to convert between polar and rectangular coordinates. These functions relate angles to the sides of a right triangle, which is why they are useful in our conversion. Specifically, we frequently use the cosine and sine functions.

Using Cosine to Find X

To find the x-coordinate from polar coordinates, we use the formula \(x = r \cos \theta\). We multiply the radial distance \(r\) by the cosine of the angle \(\theta\) to obtain the horizontal component.

Using Sine to Find Y

Similarly, the y-coordinate is found using the formula \(y = r \sin \theta\). The radial distance \(r\) is multiplied by the sine of angle \(\theta\) to compute the vertical component. For our specific example, conversions came out as follows:

• \(x = -2 \cos\left(\frac{7\pi}{6}\right)\)
• \(y = -2 \sin\left(\frac{7\pi}{6}\right)\)

These trigonometric calculations translate our polar point into easy-to-use rectangular coordinates.