Problem 35 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ (\pi, \arctan (\pi)) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{\pi}{\sqrt{1 + \pi^2}}, \frac{\pi^2}{\sqrt{1 + \pi^2}}\right)\).

Step 1: Understand the polar coordinates

The given polar coordinates are \((r, \theta) = (\pi, \arctan(\pi))\). Here, \(r = \pi\) is the radial distance from the origin, and \(\theta = \arctan(\pi)\) is the angle made with the positive x-axis.

Step 2: Use conversion formulas

The conversion from polar to rectangular coordinates is given by the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). We need to find \(x\) and \(y\) using these formulas.

Step 3: Calculate \(x\) coordinate

First, calculate \(x = r \cos \theta = \pi \cos(\arctan(\pi))\). We know that \(\cos(\arctan(a)) = \frac{1}{\sqrt{1 + a^2}}\). Thus, \(\cos(\arctan(\pi)) = \frac{1}{\sqrt{1 + \pi^2}}\). Therefore, \(x = \pi \cdot \frac{1}{\sqrt{1 + \pi^2}} = \frac{\pi}{\sqrt{1 + \pi^2}}\).

Step 4: Calculate \(y\) coordinate

Next, calculate \(y = r \sin \theta = \pi \sin(\arctan(\pi))\). We know that \(\sin(\arctan(a)) = \frac{a}{\sqrt{1 + a^2}}\). Thus, \(\sin(\arctan(\pi)) = \frac{\pi}{\sqrt{1 + \pi^2}}\). Therefore, \(y = \pi \cdot \frac{\pi}{\sqrt{1 + \pi^2}} = \frac{\pi^2}{\sqrt{1 + \pi^2}}\).

Step 5: Final rectangular coordinates

The rectangular coordinates are given by \((x, y) = \left(\frac{\pi}{\sqrt{1 + \pi^2}}, \frac{\pi^2}{\sqrt{1 + \pi^2}}\right)\).

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine and cosine are fundamental in mathematics, and they play a crucial role in converting between polar and rectangular coordinates. These functions relate the angles and sides of right triangles, allowing us to project polar information onto the rectangular plane.

• **Sine** ()} - Used to determine the vertical, or y-component of a point when converting from polar to rectangular coordinates. It represents the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine** ()} - Used to determine the horizontal, or x-component of a point during conversion. It represents the ratio of the adjacent side to the hypotenuse.

Understanding these trigonometric functions is essential because they bridge the gap between different coordinate systems, enabling seamless transitions and calculations.

Rectangular Coordinates

Rectangular coordinates are expressed as (x, y) and are used in the Cartesian plane. They describe a point's location based on its horizontal distance from the origin (x-axis) and its vertical distance (y-axis).

• **X-coordinate** - Reflects the horizontal distance from the origin to the point along the x-axis. A positive x indicates right of the origin, while a negative x suggests a leftward position.
• **Y-coordinate** - Indicates the vertical distance from the origin to the point. A positive y means above the origin, and a negative y means below it.

Rectangular coordinates are straightforward and align with basic Euclidean geometry principles, making them highly intuitive for plotting points and understanding shapes and distances.

Polar Coordinates

Polar coordinates, represented as (r, θ), determine a point's position based on a distance from the origin and an angle from the positive x-axis. They are particularly useful in scenarios where circular symmetry is prevalent, such as in fields like physics and engineering.

• **Radial Distance (r)** - The direct line from the origin to the point. A larger r means the point is farther from the origin.
• **Angle (θ)** - Measured in radians or degrees, this angle denotes how far the line from the origin to the point rotates around the origin, starting from the positive x-axis.

These coordinates are essential for understanding phenomena with rotational dynamics and are very helpful in simplifying the equations of curves that are naturally circular or spiral.

Conversion Formulas

Conversion formulas are pivotal in translating polar coordinates into rectangular coordinates and vice versa, allowing us to switch between these two systems as needed for different analytical tasks. To convert from polar to rectangular coordinates, use the following formulas:

• **X-coordinate formula**: \( x = r \cos \theta \)
• **Y-coordinate formula**: \( y = r \sin \theta \)

These equations utilize trigonometric functions to convert the radial and angular measurements from polar coordinates into the straight-line distances used in rectangular coordinates.
By mastering these formulas, you can effortlessly swap between coordinate systems, enhancing your capacity to tackle complex mathematical problems across various domains.