Problem 23 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(-1, \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

\((x, y) = \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\)

Step 1: Understanding the Problem

We are given a point in polar coordinates \((-1, \frac{5\pi}{6})\), where \(-1\) is the radius (r) and \(\frac{5\pi}{6}\) is the angle (\(\theta\)). We need to convert this to rectangular coordinates \( (x, y) \) using the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 2: Using the Formula for x

Apply the formula for x: \(\begin{align*}x &= r \cos(\theta) \ &= -1 \cos\left(\frac{5\pi}{6}\right).\end{align*} \)Use the unit circle to find that \(\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}\). Thus, \(x = -1 \times \left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2}\).

Step 3: Using the Formula for y

Apply the formula for y: \(y = r \sin(\theta) = -1 \sin\left(\frac{5\pi}{6}\right) \).Using the unit circle, \(\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}\). Therefore, \(y = -1 \times \frac{1}{2} = -\frac{1}{2}\).

Step 4: Combine Results

Combine the results from the previous steps to express the rectangular coordinates as \((x, y) = \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\).

Key Concepts

Polar Coordinates

Polar coordinates offer a different way of expressing the location of a point in a plane. In this system, each point is determined by a distance from a reference point and an angle from a reference direction. The two components of a polar coordinate are:

• **Radius ( ef r ef r)**: This is the distance from the origin to the point.
• **Angle ( ef θ ef θ)**: This is the angle between the positive x-axis and the line connecting the origin with the point. It is usually measured in radians.

To understand polar coordinates, imagine using a compass to draw circles: the radius tells you how far from the center, and the angle tells you in which direction. This system is especially useful in scenarios involving circular paths or when working with periodic functions.
The conversion to rectangular coordinates involves calculating the horizontal and vertical components of this point on a typical x-y plane.

Rectangular Coordinates

In contrast to polar coordinates, rectangular coordinates describe a point using horizontal and vertical positions from a fixed reference. This system uses two values, known as x and y, which represent distances:

• **x-coordinate (horizontal distance):** Shows how far the point is along the x-axis, starting from the origin.
• **y-coordinate (vertical distance):** Shows how far up or down the point is along the y-axis, from the starting point of the origin.

Together, the x and y values give the exact position of the point within a flat plane. Rectangular coordinates are more common and easier to use when working with straight lines and rectangular shapes. The challenge arises when converting them from or to polar forms, which requires using trigonometric functions.

Unit Circle

The **unit circle** is a fundamental concept in trigonometry and unit used to understand angles and lengths. It is a circle with a radius of 1, centered at the origin of a coordinate system (the point ef(0,0) ef in the ef x-y ef plane).
The importance of the unit circle lies in its ability to simplify calculations involving trigonometric functions. The angle on the unit circle corresponds to the angle of a ray emanating from the origin. Each point on the circle can be associated with an angle that measures its position:

• The x-coordinate corresponds to ef ext{cos}( heta) ef of that angle.
• The y-coordinate corresponds to ef ext{sin}( heta) ef of that angle.

The unit circle helps to easily visualize and retrieve exact values for common angles ef like rac{5 heta}{6} ef, precisely because these angles correspond to points symmetrically arranged around the circle.

Trigonometric Functions

Trigonometric functions play a crucial role in converting between polar and rectangular coordinates. These functions relate angles and sides within right triangles but are also applied to analyze circles, like the positions on the unit circle. The primary trigonometric functions are:

• **Sine ( ef ext{sin} ef):** Represents the y-coordinate or "opposite over hypotenuse" in a triangle.
• **Cosine ( ef ext{cos} ef):** Represents the x-coordinate or "adjacent over hypotenuse."

When converting from polar coordinates ef(r, heta) ef to rectangular coordinates ef(x, y) ef, the formulas are:

• ef x = r ext{cos}( heta) ef
• ef y = r ext{sin}( heta) ef

These calculations rely heavily on the well-known values from the unit circle, allowing us to determine ef ext{cos}(rac{5 heta}{6}) = -rac{ heta heta{3}}{2} ef and eft ext{sin}(rac{5 heta}{6}) = rac{1}{2} ef directly. This is how we can adeptly switch between the two coordinate systems.