Problem 34 - College Trigonometry

Question

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

Step-by-Step Solution

Answer

The rectangular coordinates are \((-\frac{2}{9}, -\frac{4\sqrt{2}}{9})\).

Step 1: Understand the Polar Coordinates

The polar coordinates given are \( (r, \theta) \) which are \( \left( \frac{2}{3}, \pi + \arctan (2 \sqrt{2}) \right) \). In polar coordinates, \( r \) is the radius from the origin, and \( \theta \) is the angle from the positive x-axis.

Step 2: Formula for Conversion

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas:- \( x = r \cos \theta \)- \( y = r \sin \theta \).

Step 3: Calculate \( \theta \)

The angle \( \theta = \pi + \arctan (2 \sqrt{2}) \). Use the addition angle trigonometric identities and recognize the significance of \( \pi \) which shifts the angle by 180 degrees.

Step 4: Find \( \cos \theta \) and \( \sin \theta \)

Since \( \theta = \pi + \arctan(2 \sqrt{2}) \), use:- \( \cos(\pi + \arctan(2\sqrt{2})) = -\cos(\arctan(2\sqrt{2})) \)- \( \sin(\pi + \arctan(2\sqrt{2})) = -\sin(\arctan(2\sqrt{2})) \).Since \( \tan \theta = 2\sqrt{2} \), we use the identity \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). Here, opposite = \(2\sqrt{2}\) and adjacent = 1, so the hypotenuse is \(\sqrt{(2\sqrt{2})^2 + 1^2} = 3\).

Step 5: Determine \( \cos(\arctan(2\sqrt{2})) \) and \( \sin(\arctan(2\sqrt{2})) \)

Using the right triangle, \( \cos(\arctan(2\sqrt{2})) = \frac{1}{3} \) and \( \sin(\arctan(2\sqrt{2})) = \frac{2\sqrt{2}}{3} \). So,- \( \cos(\pi + \arctan(2\sqrt{2})) = -\frac{1}{3} \) - \( \sin(\pi + \arctan(2\sqrt{2})) = -\frac{2\sqrt{2}}{3} \).

Step 6: Calculate Rectangular Coordinates

Apply the polar to rectangular formulas:- \(x = r \cos \theta = \frac{2}{3} \times -\frac{1}{3} = -\frac{2}{9} \).- \(y = r \sin \theta = \frac{2}{3} \times -\frac{2\sqrt{2}}{3} = -\frac{4\sqrt{2}}{9} \).

Step 7: Conclusion

The rectangular coordinates corresponding to the given polar coordinates are \( \left(-\frac{2}{9}, -\frac{4\sqrt{2}}{9}\right) \).

Key Concepts

Polar Coordinates

Polar coordinates are a unique way to describe a point's location on a plane. Typically denoted as \((r, \theta)\), polar coordinates use two components:

• Radius \(r\): The straight-line distance from the origin to the point.
• Angle \(\theta\): Measured in radians, representing the angle from the positive x-axis to the line extending to the point.

In the given exercise, the polar coordinates are \(\left( \frac{2}{3}, \pi + \arctan(2 \sqrt{2}) \right)\). The radius is \(\frac{2}{3}\) which is measured from the origin, and the angle \(\theta\) is a cumulative angle resulting from \(\pi\) and \(\arctan(2 \sqrt{2})\).Understanding polar coordinates makes it possible to easily convert them into the more familiar rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point's position in terms of its horizontal and vertical components. They are represented as \((x, y)\), where

• x: The horizontal distance from the origin.
• y: The vertical distance from the origin.

In a coordinate plane, every point can also be expressed using rectangular coordinates, and converting from polar coordinates to rectangular coordinates is a common technique in mathematics.The conversion process involves using the polar coordinates formula \[\begin{align*}x &= r \cos \theta \y &= r \sin \theta\end{align*}\]This forms the basis for converting from polar to rectangular coordinates. For this exercise, these formulas help transform the point to \(\left(-\frac{2}{9}, -\frac{4\sqrt{2}}{9}\right)\).

Trigonometric Identities

Trigonometric identities are crucial tools that make conversions like these possible. In particular, the addition formulas can simplify the expressions of trigonometric functions. Understanding how these identities work will enhance your math skills. Here are the fundamental addition identities used in the exercise:

• Cosine Addition Identity: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
• Sine Addition Identity: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]

For the given problem, to adjust for the angle shift by \(\pi\), use these cosine and sine identities in the reverse direction. Knowing these identities not only helps in performing conversions but also in solving many kinds of trigonometric equations effectively.

Angle Conversion

Angle conversion refers to adjusting angles expressed in different terms. This is essential when working with trigonometric functions and transformations. In this exercise, the angle \(\theta\) within the polar coordinates involves two parts:

• \(\pi\): Represents a rotation mathematically, equivalent to 180 degrees, effectively flipping the angle's direction.
• \(\arctan(2\sqrt{2})\): Details a specific angle whose tangent is \(2\sqrt{2}\), giving the precise direction to consider post-rotation.

The task becomes how these two angles coexist to affect the trigonometric computations. Instead of evaluating \(\pi + \theta\) directly, one uses the identities to compute the adjusted cosine and sine values. It's a clever use of math principles to achieve accurate coordinates in a different space.