Problem 22 - Precalculus

Question

Convert each point to exact rectangular coordinates. $$\left(2, \frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify Initial Polar Coordinates

The given point in polar coordinates is \((r, \theta) = (2, \frac{3\pi}{4})\). Here, \(r = 2\) is the radius and \(\theta = \frac{3\pi}{4}\) is the angle in radians.

Step 2: Apply Conversion Formulas

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

Step 3: Calculate Rectangular Coordinate x

Substitute \(r = 2\) and \(\theta = \frac{3\pi}{4}\) into the formula for \(x\): \[x = 2 \cos\left(\frac{3\pi}{4}\right)\]Since \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \[x = 2 \times \left(-\frac{\sqrt{2}}{2}\right) = -\sqrt{2}\].

Step 4: Calculate Rectangular Coordinate y

Substitute \(r = 2\) and \(\theta = \frac{3\pi}{4}\) into the formula for \(y\): \[y = 2 \sin\left(\frac{3\pi}{4}\right)\]Since \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\), \[y = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\].

Step 5: Write the Final Answer

The rectangular coordinates corresponding to the polar coordinates \((2, \frac{3\pi}{4})\) are \((-\sqrt{2}, \sqrt{2})\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to specify the unique location of a point on a plane using two dimensions. These coordinates consist of two value pairs \( x \) and \( y \), which represent the horizontal and vertical distances from the origin, respectively.

They are structured in the form \( (x, y) \), where \( x \) is the coordinate along the horizontal x-axis, and \( y \) is the coordinate along the vertical y-axis. This system is very intuitive because it is extensively used in algebra, geometry, and various real-world applications like maps and designs.

• x-coordinate: Measures how far right (positive) or left (negative) the point is from the origin (0,0).
• y-coordinate: Measures how far up (positive) or down (negative) the point is from the origin.

Understanding how to navigate between rectangular coordinates and other systems, like polar coordinates, is essential for tackling various mathematical and engineering problems.

Polar Coordinates

Polar coordinates provide an alternative way to describe points in a plane. They are particularly useful for problems involving circular and spiral patterns. Instead of using horizontal and vertical distances like the Cartesian system, polar coordinates define a point’s location through:

• Radius (r): The distance from the point to the origin.
• Angle (θ): Measured in radians, it describes the direction from the origin to the point.

Polar coordinates are represented in the form \( (r, \theta) \). The angle is measured from the positive x-axis in a counterclockwise direction. Conversion between polar and rectangular coordinates requires a good understanding of trigonometry since it involves using trigonometric functions to translate the radius and angle into \( x \) and \( y \) coordinates.

Some common uses of polar coordinates are in physics and engineering to model systems with rotational symmetry or in situations where directions are involved.

Trigonometric Functions

Trigonometric functions play a critical role in converting between polar and rectangular coordinates. The main trigonometric functions involved in this process are sine and cosine.

• Cosine (cos): Represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is used to calculate how much the radius projection reaches along the x-axis when converting polar coordinates to rectangular form.
• Sine (sin): Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. It tells us how much the radius projects onto the y-axis when converting from polar to rectangular coordinates.

Using the formulas \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), you can find the equivalent rectangular coordinates. It’s important to note that angles in polar coordinates are typically measured in radians. For example, in this exercise, we converted \( \left(2, \frac{3 \pi}{4} \right) \) to \( (x, y) \) by using these trigonometric functions, yielding \( x = -\sqrt{2} \) and \( y = \sqrt{2} \). Understanding these functions is crucial for solving problems that involve any form of oscillations, rotations, or periodic motions in various fields.