Problem 65 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (5,5 \sqrt{3}) $$

Step-by-Step Solution

Answer

(10, \( \frac{\pi}{3} \))

Step 1: Understand the relationship between rectangular and polar coordinates

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), use the equations: \[ r = \sqrt{x^2 + y^2} \] and \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Step 2: Calculate the radial coordinate (r)

For the given point \( (5, 5\sqrt{3}) \): \[ r = \sqrt{(5)^2 + (5\sqrt{3})^2} = \sqrt{25 + 75} = \sqrt{100} = 10 \]

Step 3: Calculate the angular coordinate (\(\theta\))

For the given point \( (5, 5\sqrt{3}) \): \[ \theta = \tan^{-1}\left(\frac{5\sqrt{3}}{5}\right) = \tan^{-1}(\sqrt{3}) \] \(\theta\) corresponding to \(\sqrt{3}\) is \(\frac{\pi}{3}\).

Step 4: Combine the results

The polar coordinates are: \( r = 10, \theta = \frac{\pi}{3} \). So, \( (10, \frac{\pi}{3}) \)

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are expressed as \(x, y\). These coordinates describe positions on a plane by specifying how far a point is from the two perpendicular, horizontal (x-axis) and vertical (y-axis) lines.

In our problem, the given rectangular coordinates are (5, 5\(\sqrt{3}\)).

• The first number (5) represents the position along the x-axis.
• The second number \(5\sqrt{3}\) represents the position along the y-axis.

Knowing these positions helps us locate the exact point on a two-dimensional plane but can sometimes be cumbersome for understanding certain relationships or calculating distances and angles, this is where polar coordinates come in handy.

Polar Coordinates

Polar coordinates describe the position of a point using the distance from a reference point (usually the origin) and an angle from a reference direction (usually the positive x-axis). Polar coordinates are expressed as \(r, \theta\), where:

• \( r \) is the radial distance from the origin to the point
• \( \theta \) is the angle between the positive x-axis and the line connecting the origin to the point

For example, transforming our point \(5, 5\sqrt{3}\) into polar coordinates, we find:

• \( r \), calculated via \( \sqrt{x^2 + y^2} \)
• \( \theta \), determined by \( \arctan\left(\frac{y}{x}\right) \)

This gives us the coordinates (10, \(\frac{\pi}{3}\)). The use of polar coordinates is particularly useful in trigonometry and scenarios involving circular or rotational motion.

Coordinate Conversion

Converting rectangular coordinates to polar coordinates requires understanding key equations:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \arctan\left(\frac{y}{x}\right) \)

These formulas stem from the Pythagorean theorem and trigonometric functions.

For instance, to convert the point (5, 5\(\sqrt{3}\)) to polar coordinates, we:

• Calculate \( r \) as \( \sqrt{5^2 + (5\sqrt{3})^2} = 10 \)
• Calculate \( \theta \) as \( \arctan\left(\frac{5\sqrt{3}}{5}\right) = \arctan(\sqrt{3}) = \frac{\pi}{3} \)

This straightforward conversion is essential for various applications in trigonometry, physics, and engineering.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships between side lengths and angles of triangles. In coordinate conversion:

• The Pythagorean theorem is used to find the distance (r).
• The tangent function and its inverse help to determine the angle \( \theta \).

For our exercise, the radial distance is found using the formula derived from the Pythagorean theorem:

• \( r = \sqrt{x^2 + y^2} \)

The angle \( \theta \) is then found using the arctangent function:

• \( \theta = \arctan\left(\frac{y}{x}\right) \)

These trigonometric principles are foundational to converting rectangular coordinates into polar coordinates and vice versa.