Problem 21 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(\frac{3}{5}, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((0, \frac{3}{5})\).

Step 1: Understand the Problem

We need to convert the polar coordinates given as \( \left(\frac{3}{5}, \frac{\pi}{2}\right) \) into rectangular coordinates. In polar coordinates, the first value represents the radius \( r \), and the second value represents the angle \( \theta \) from the positive x-axis.

Step 2: Recall the Conversion Formulas

The conversion from polar to rectangular coordinates is done using the formulas: \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \). We will use these to find the \( x \) and \( y \) coordinates.

Step 3: Calculate the x-coordinate

Substitute \( r = \frac{3}{5} \) and \( \theta = \frac{\pi}{2} \) into the equation for \( x \): \( x = \frac{3}{5} \cdot \cos\left(\frac{\pi}{2}\right) \). We know \( \cos\left(\frac{\pi}{2}\right) = 0 \), so \( x = \frac{3}{5} \cdot 0 = 0 \).

Step 4: Calculate the y-coordinate

Substitute \( r = \frac{3}{5} \) and \( \theta = \frac{\pi}{2} \) into the equation for \( y \): \( y = \frac{3}{5} \cdot \sin\left(\frac{\pi}{2}\right) \). We know \( \sin\left(\frac{\pi}{2}\right) = 1 \), so \( y = \frac{3}{5} \cdot 1 = \frac{3}{5} \).

Step 5: Write the Rectangular Coordinates

The rectangular coordinates are \( (x, y) = (0, \frac{3}{5}) \). Therefore, the polar coordinates \( \left(\frac{3}{5}, \frac{\pi}{2}\right) \) convert to rectangular coordinates \( (0, \frac{3}{5}) \).

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way of expressing a point's position using a distance and an angle. This system is particularly useful in scenarios where direction and magnitude from a particular point are crucial.

• The **radius** \( r \) represents the distance from the origin (or the pole) to the point.
• The **angle** \( \theta \) is the direction from the positive horizontal axis.

A common scenario where polar coordinates are used is in representing the position of rotating objects, like a clock's hands, or in navigation and radar systems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, provide a straightforward way to describe a point's location using two intersecting lines: the x-axis and the y-axis. Each point on a plane can be uniquely identified by an \( x \) and \( y \) value.

• The **x-coordinate** determines how far along the horizontal axis the point lies.
• The **y-coordinate** shows the position along the vertical axis.

This system is used in various domains such as computer graphics, engineering, physics, and more, due to its simplicity and effectiveness in representing geometrical shapes and other computational models.

Conversion Formulas

Converting from polar to rectangular coordinates requires understanding of the mathematical relationship between these systems. The conversion relies on simple yet essential formulas derived from trigonometric principles.

• The formula for the x-coordinate: \( x = r \cdot \cos(\theta) \)
• The formula for the y-coordinate: \( y = r \cdot \sin(\theta) \)

These formulas come from the relationships between the sides and angles of a right triangle, where \( r \) is the hypotenuse, making this approach intuitive for the conversion process.

Trigonometric Functions

Trigonometric functions like sine and cosine are central to the conversion between polar and rectangular coordinates. These functions relate angles to side lengths in right triangles.

• **Sine (\( \sin \)):** Opposite side over hypotenuse. Used to find the y-coordinate in polar-to-rectangular conversions.
• **Cosine (\( \cos \)):** Adjacent side over hypotenuse. Crucial for calculating the x-coordinate.

Understanding these functions can help grasp the polar-to-rectangular coordinate conversions. In our example, knowing that \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \) makes it easier to see how the math directly translates coordinates from one system to another.