Problem 34 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (\sqrt{3},-5 \pi / 3) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \( \left( \frac{\sqrt{3}}{2}, -\frac{3}{2} \right) \).

Step 1: Understand Polar to Rectangular Conversion

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius or the distance from the origin, and \(\theta\) is the angle in radians from the positive x-axis. To convert these to rectangular coordinates \((x, y)\), we use the formulas:\[ x = r \cos \theta \]\[ y = r \sin \theta \]

Step 2: Apply the Formulas

Given the polar coordinates \((\sqrt{3}, -\frac{5\pi}{3})\), we can find the rectangular coordinates using the formulas:\[ x = \sqrt{3} \cos\left(-\frac{5\pi}{3}\right) \]\[ y = \sqrt{3} \sin\left(-\frac{5\pi}{3}\right) \]

Step 3: Calculate \(\cos(-\frac{5\pi}{3})\) and \(\sin(-\frac{5\pi}{3})\)

The angle \(-\frac{5\pi}{3}\) is equivalent to \( \frac{\pi}{3} \) since adding \(2\pi\) gives the same position on the unit circle. From the unit circle, we know:\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Step 4: Substitute Values into the Formulas

Now, substitute the calculated values of \(\cos\) and \(\sin\) into the \(x\) and \(y\) formulas:\[ x = \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \]\[ y = \sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{3}{2} \]

Step 5: Determine the Signs of \(x\) and \(y\)

Since the angle is \(-\frac{5\pi}{3}\), which corresponds to \(\frac{\pi}{3}\) in the fourth quadrant, the \(x\)-coordinate is positive, and the \(y\)-coordinate is negative. Therefore, the rectangular coordinates are:\[ \left( \frac{\sqrt{3}}{2}, -\frac{3}{2} \right) \]

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationship between angles and side lengths in triangles. It plays a central role in converting polar coordinates to rectangular coordinates because it involves the use of sine and cosine functions. These trigonometric functions help find the

• Horizontal displacement, known as the cosine,
• Vertical displacement, known as the sine.

In a polar to rectangular conversion, the sine and cosine of an angle \(\theta\) determine the \(x\) and \(y\) coordinates of a point based on the \(r\), or radius. The formulas used are straightforward:
\[ x = r \cos \theta \]\[ y = r \sin \theta \]These functions relate the angle's

• Distance along the x-axis (cosine),
• Distance along the y-axis (sine),

helping to seamlessly convert angles and distances from the polar to the familiar rectangular coordinate system.
Understanding trigonometry is key to mastering many concepts in mathematics, physics, and engineering.

Rectangular Coordinates

Rectangular coordinates are part of the Cartesian coordinate system, which is widely used to specify locations on a flat plane. Each point in this system has two coordinates:

• \(x\), the horizontal position, and
• \(y\), the vertical position.

This system is likely already familiar as it forms the basis for graphing functions and solving geometric problems.
For conversion from polar coordinates, the goal is to find these \(x\) and \(y\) coordinates given the polar radius \(r\) and angle \(\theta\).
In the given exercise, by converting

• \( \sqrt{3} \cos(-\frac{5\pi}{3}) \) to calculate \(x\), and
• \( \sqrt{3} \sin(-\frac{5\pi}{3}) \) to calculate \(y\),

the point

• (\(\frac{\sqrt{3}}{2}, -\frac{3}{2}\))

is obtained in rectangular coordinates. It utilizes simple trigonometric evaluations on a familiar number plane.

Unit Circle

The unit circle is a foundational concept in trigonometry that provides a geometric way of understanding angles and their corresponding sine and cosine values. It is a circle with a radius of one, centered at the origin of the Cartesian plane.

• Angles are measured in radians, and circle's total circumference is \(2\pi\) radians.
• Any point on the unit circle can be expressed as \( (\cos \theta, \sin \theta) \).

In this exercise, the unit circle helps calculate the trigonometric functions for the given angle \(\theta = -\frac{5\pi}{3}\).
By visualizing the unit circle, we see that \(\theta\) places us in the fourth quadrant, where

• The x-coordinate (cosine) is positive
• The y-coordinate (sine) is negative.

Understanding these properties allows us to accurately convert polar coordinates to rectangular coordinates with confidence.

Polar Coordinates

Polar coordinates describe the location of a point in terms of

• \(r\) - the distance from the origin, and
• \(\theta\) - the angle from a direction known as the positive x-axis.

This system is particularly useful in cases involving symmetry and curves, such as

• circles,
• spirals,

because it naturally describes radial distances and angular displacement.
In the task's context, the polar coordinates \(\left(\sqrt{3}, -\frac{5\pi}{3}\right)\) means:

• A point \(\sqrt{3}\) units away from the origin.
• Positioned at an angle of \(-\frac{5\pi}{3}\).

Conversion from polar to rectangular coordinates involves finding corresponding \(x,.\)and \(y\) values on a plane. This conversion is done using the basic trigonometric principles described earlier.