Problem 47 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(6, \frac{5 \pi}{6}\right) $$

Step-by-Step Solution

Answer

Rectangular coordinates are \( \left(-3 fabs 3, 3\right) \).

Step 1 Title - Understand Polar Coordinates

Polar coordinates are given as \( r, \theta \). Here, \( r \) represents the radius or the distance from the origin, and \( \theta \) represents the angle in radians measured counterclockwise from the positive x-axis.

Step 2 Title - Identify Given Values

From the given polar coordinates \( \left(6, \frac{5 \pi}{6}\right) \), we see that \( r = 6 \) and \( \theta = \frac{5 \pi}{6} \).

Step 3 Title - Convert to Rectangular Coordinates

To convert from polar coordinates to rectangular coordinates, use the formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).

Step 4 - Calculate x-coordinate

Substitute \( r = 6 \) and \( \theta = \frac{5 \pi}{6} \) into the x-coordinate formula: \( x = 6 \cos\left(\frac{5 \pi}{6}\right) \). Since \( \cos\left(\frac{5 \pi}{6}\right) = -\frac{\fabs{3}}{2} \), we get \( x = 6 \left(-\frac{\fabs{3}}{2}\right) = -3 fabs 3\).

Step 5 Title - Calculate y-coordinate

Now, substitute \( r = 6 \) and \( \theta = \frac{5 \pi}{6} \) into the y-coordinate formula: \( y = 6 \sin\left(\frac{5 \pi}{6}\right) \). Since \( \sin\left(\frac{5 \pi}{6}\right) = \frac{1}{2} \), we get \( y = 6 \frac{1}{2} = 3 \).

Step 6 Title - Write Rectangular Coordinates

Combine the x and y values to get the rectangular coordinates: \( \left(-3 fabs 3, 3\right) \).

Key Concepts

Polar Coordinates

Polar coordinates are a different way to represent points in a plane using a radius and an angle instead of the traditional x and y methods. In this system, any point is described by two values:

• **_r_**, the distance from the origin (0,0) to the point
• **_θ_**, the angle from the positive x-axis to the point, measured in radians

For example, the polar coordinates \((6, \frac{5 \pi}{6})\) means that the point is 6 units away from the origin and located at an angle of \frac{5 \pi}{6} radians. This angle is approximately 150 degrees.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the traditional way of describing a point in a plane using an x and y value. Here,

• **_x_**, represents the horizontal distance from the origin
• **_y_**, represents the vertical distance from the origin

Suppose we have a point P with rectangular coordinates \((-3 \sqrt{3}, 3)\). It means that from the origin, you move 3\sqrt{3} units to the left along the x-axis and 3 units up along the y-axis.

Coordinate Conversion

Converting between polar coordinates and rectangular coordinates involves using some basic trigonometric functions:

• To convert from polar to rectangular:
• **_x_ = r \cos(\theta)**
• **_y_ = r \sin(\theta)**

Let's use the given polar coordinates \(6, \frac{5 \pi}{6})\):

• First, identify _r_ and _θ_. Here, _r_ is 6 and _θ_ is \frac{5 \pi}{6}
• Next, calculate the x-coordinate using x = 6 \cos(\frac{5 \pi}{6})

Since \cos(\frac{5 \pi}{6}) = -\sqrt{3}/2, we get:
x = 6 (-\sqrt{3}/2) = -3\sqrt{3}.
Now, calculate the y-coordinate using: y = 6 \sin(\frac{5 \pi}{6}). Since \sin(\frac{5 \pi}{6}) = 1/2, we get:
y = 6 (1/2) = 3.
Therefore, the rectangular coordinates are: ( -3 \sqrt{3}, 3).
This systematic approach makes the conversion straightforward and logical.