Problem 30 - Precalculus : Building Concepts and Connections

Question

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(-2,2 \sqrt{3})$$

Step-by-Step Solution

Answer

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

Step 1: Calculating r

First, calculate the distance \( r \) from the origin to the point. This is done using Pythagorean theorem which gives us \( r = \sqrt{x^2 + y^2} \). When we input our coordinates we get \( r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4 \). So, \( r = 4 \)

Step 2: Calculating \(\theta\)

To calculate \(\theta\), we use the formula \( \tan(\theta) = \frac{y}{x} \). Then we use the arctan or tan inverse to find \(\theta\). However, we have to consider the quadrant of the point here. The point (-2, 2\( \sqrt{3} \)) is in the second quadrant, therefore \(\theta\) will lie between \(\pi / 2\) and \(\pi\). Calculating \(\theta\), we find \( \tan(\theta) = \frac{2\sqrt{3}}{-2} = -\sqrt{3} \). Now, we know that arctan(\(-\sqrt{3}\)) is \(-\pi / 3\). But since the point is in the second quadrant, we add \(\pi\) to get the actual \(\theta\). Hence, \(\theta = -\pi / 3 + \pi = 2\pi / 3 \).

Step 3: Final Answer

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

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly when dealing with right-angled triangles. It states that for any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is expressed in the formula:
\[ c^2 = a^2 + b^2 \]
where \( c \) represents the length of the hypotenuse, and \( a \) and \( b \) represent the lengths of the triangle's other two sides.

In the context of converting rectangular coordinates to polar coordinates, the Pythagorean theorem helps us determine the radial distance \( r \) from the origin to the point. Here, \( r \) serves as the hypotenuse of a right triangle formed by drawing a perpendicular from the point to the X-axis. The two legs of this triangle are the X and Y coordinates of the point, and thus the theorem is used as follows:\[ r = \sqrt{x^2 + y^2} \]
This step is crucial because it provides the magnitude of the polar coordinate, which tells us how far away the point is from the origin in a direct line.

Polar Coordinates

Polar coordinates offer an alternative way to represent the position of a point in a plane, using a distance and an angle rather than the traditional X and Y coordinates. In polar coordinates, a point is identified by two values:

• The radial distance \( r \) from the origin (also known as the pole), which is the length of a straight line drawn from the origin to the point.
• The angle \( \theta \) (theta), which is the counterclockwise rotation required to align the positive X-axis with a line drawn from the origin to the point. This angle is measured in radians.

Both \( r \) and \( \theta \) are needed to precisely locate a point on a plane using this system. The conversion from rectangular to polar coordinates involves calculating these two values based on the given X and Y coordinates.

Trigonometric Functions

Trigonometric functions are the bridge between angular and spatial relationships in a plane, and they play a crucial role in the conversion of rectangular coordinates to polar coordinates. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which relate the angles of a right triangle to the ratios of two of its sides.

For a right triangle with angle \( \theta \), the functions are defined as:

• \( \sin(\theta) = \frac{{\text{{opposite side}}}}{{\text{{hypotenuse}}}} \)
• \( \cos(\theta) = \frac{{\text{{adjacent side}}}}{{\text{{hypotenuse}}}} \)
• \( \tan(\theta) = \frac{{\text{{opposite side}}}}{{\text{{adjacent side}}}} \)

In conversion problems, the tangent function is often used to find the angle \( \theta \) since it directly relates the Y coordinate (opposite side) to the X coordinate (adjacent side). Calculating this ratio and then finding the angle that has this tangent value using the inverse tangent function (arctan), gives us the measure of \( \theta \) in radians.

Quadrant Consideration

The concept of quadrants is essential when working with polar coordinates because the angle \( \theta \) can take on different values depending on the quadrant in which the point is located. The coordinate plane is divided into four quadrants by the X-axis and Y-axis, labelled counter-clockwise as I, II, III, and IV.

Knowing the quadrant helps determine the correct direction of \( \theta \):

• In Quadrant I, \( 0 \leq \theta < \frac{\pi}{2} \).
• In Quadrant II, \( \frac{\pi}{2} \leq \theta < \pi \).
• In Quadrant III, \( \pi \leq \theta < \frac{3\pi}{2} \).
• In Quadrant IV, \( \frac{3\pi}{2} \leq \theta < 2\pi \).

When converting coordinates, if the result of arctan gives an angle not corresponding to the correct quadrant, it's necessary to adjust the angle by adding or subtracting \( \pi \) (since \( \tan(\theta + n\pi) = \tan(\theta) \) for any integer \( n \)). This ensures that the angle reflects the true direction from the origin to the point, and satisfies the condition \( 0 \leq \theta < 2\pi \).