Problem 24 - Contemporary Precalculus

Question

Convert the rectangular coordinates to polar coordinates. $$(\sqrt{2},-\sqrt{2})$$

Step-by-Step Solution

Answer

Question: Convert the rectangular coordinates \((\sqrt{2}, -\sqrt{2})\) to polar coordinates. Answer: The polar coordinates are \((2, 7\pi/4)\).

Step 1: Find r (the distance from the origin to the point)

To find r, use the Pythagorean theorem: \(r = \sqrt{x^2 + y^2}\). For the given point \((\sqrt{2}, -\sqrt{2})\), x = \(\sqrt{2}\), and y = \(-\sqrt{2}\). $$r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2$$

Step 2: Find θ (the angle formed between the positive x-axis and the point)

To find the angle θ, use the tangent function: \(\tan(\theta) = \frac{y}{x}\). θ can be found using the inverse tangent function, \(\tan^{-1}(\frac{y}{x})\): $$\theta = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(-1)$$ In order to determine which quadrant the angle should be placed, we look at the signs of x and y coordinates. Since x is positive and y is negative, the angle should lie in the 4th quadrant. Recall that \(\tan^{-1}(-1)\) is associated with a \(45^\circ\) angle in the 4th quadrant, which (in radians) is equal to \(7\pi/4\). So, the polar coordinates of the given point are \((2, 7\pi/4)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, provide a system to pinpoint the location of a point in a plane using an ordered pair \(x, y\). This system is named after René Descartes, who formalized its concepts. In a 2D plane, the x-coordinate tells us how far to move horizontally from the origin (0,0). The y-coordinate indicates how far to move vertically. For example, the point \(\sqrt{2}, -\sqrt{2}\) means you move \sqrt{2}\ units to the right along the x-axis (since it is positive) and then \sqrt{2}\ units down along the y-axis (since it is negative). This forms a distinct point in the coordinate plane. Using rectangular coordinates is essential for plotting points, graphing functions, and solving geometric problems.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that relates the sides of a right-angled triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is the sum of the squares of the other two sides. In mathematical terms, for a right-angled triangle with sides a, b, and hypotenuse c, the equation is given by: \\[ c^2 = a^2 + b^2 \] When converting rectangular coordinates \(x, y\) to polar coordinates, the Pythagorean theorem is used to find the distance r from the origin to the point: \\[ r = \sqrt{x^2 + y^2} \] In our example, finding the distance for \(\sqrt{2}, -\sqrt{2}\) results in: \\[ r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \] This calculation tells us how far the point is from the origin, effectively giving it a magnitude when using polar coordinates.

Inverse Tangent Function

The inverse tangent function, also called arctangent and denoted as \(\tan^{-1}(x)\), is used to find an angle whose tangent is x. It is crucial for converting Cartesian coordinates to polar coordinates when the angle \(\theta\) needs to be determined. Recall the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. For the point \(\sqrt{2}, -\sqrt{2}\), the tangent can be calculated as: \\[ \tan(\theta) = \frac{y}{x} = \frac{-\sqrt{2}}{\sqrt{2}} = -1 \] Then, using the inverse tangent function, we find the angle: \\[ \theta = \tan^{-1}(-1) \] It's important to remember the resulting angle from the inverse tangent will depend on the quadrant in which the point lies. In standard conventions, \(\tan^{-1}(-1)\) corresponds to \(-\frac{\pi}{4}\) in the 4th quadrant, equivalent to \(\frac{7\pi}{4}\) in positive measure.

Quadrant Analysis

Quadrant analysis is used to determine the correct angle \(\theta\) for a point in polar coordinates by analyzing the signs of the x and y values in rectangular coordinates. The coordinate plane is divided into four quadrants:

• 1st Quadrant: Both x and y are positive.
• 2nd Quadrant: x is negative, y is positive.
• 3rd Quadrant: Both x and y are negative.
• 4th Quadrant: x is positive, y is negative.

In our given exercise, the point \(\sqrt{2}, -\sqrt{2}\) is evaluated. Here, x is positive while y is negative, placing it in the 4th quadrant.This information is vital as it impacts how we interpret \(\tan^{-1}(\frac{y}{x})\). For points in the 4th quadrant, angles are measured in a clockwise direction and can be expressed in radians, such as \(\frac{7\pi}{4}\) for this example.