Problem 43 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$ (2,-2 \sqrt{3}) $$

Step-by-Step Solution

Answer

The polar coordinates of the point (2, -2\sqrt{3}) are (4, 5\pi/3).

Step 1: Calculate r

The formula for r is \(r = \sqrt{x^2 + y^2}\). By substituting \(x = 2\) and \(y = -2\sqrt{3}\), you get \(r = \sqrt{(2)^2+(-2\sqrt{3})^2}=4\). Therefore, \(r = 4\).

Step 2: Determine quadrant

The given coordinates (2, -2\sqrt{3}) lie in the fourth quadrant because x>0 and y<0. Hence, the formula to be used for \(\theta\) will be \(\theta = \arctan(y/x)\).

Step 3: Calculate θ in radians

Use the formula \(\theta = \arctan(y/x)\) to find θ. Substituting \(x = 2\) and \(y = -2\sqrt{3}\), you get \(\theta = \arctan(-2\sqrt{3}/2) = -\pi/3\). However, -\pi/3 results in a negative angle which points in the third quadrant. As our coordinate is in the fourth quadrant, we need to add 2pi to the result to get it to point in the right direction. Therefore, \(\theta = -\pi/3 + 2\pi = 5\pi/3\).

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are defined by a pair \(x, y\). They represent a point's location on a two-dimensional plane. This coordinate system divides the plane into grid-like sections using horizontal (x-axis) and vertical (y-axis) lines.

Here's how it works:

• The x-coordinate determines the position along the horizontal axis.
• The y-coordinate specifies the position along the vertical axis.

For example, the point (2, -2√3) has an x-value of 2, placing it 2 units to the right of the origin. The y-value is -2√3, placing it \(-2\sqrt{3}\) units downward. This coordinate system is commonly used in mathematical problems for its simplicity and direct visualization.

Exploring Quadrants

A two-dimensional plane is divided into four quadrants. Each is determined by a combination of positive or negative values of x and y. Here's what you need to know:

• First Quadrant: x > 0, y > 0
• Second Quadrant: x < 0, y > 0
• Third Quadrant: x < 0, y < 0
• Fourth Quadrant: x > 0, y < 0

For the point (2, -2√3), both x is positive and y is negative. This combination places the point in the fourth quadrant. Identifying the correct quadrant is essential in converting rectangular coordinates to polar coordinates, as it determines the sign and magnitude of the angle \(\theta\).

Angle in Radians Explained

Angles can be expressed in degrees or radians, but in mathematics, radians are often preferred because they simplify many formulas. Radians measure the angle as the length of the arc divided by the radius. Here's a quick recap:

• One complete revolution around a circle is \(2\pi\) radians.
• A semicircle measures \(\pi\) radians, which is equivalent to 180 degrees.
• Hence, \(\pi\over 3\) equals 60 degrees, and so forth.

The conversion to radians allows for straightforward calculation in trigonometry and calculus. In the exercise above, converting \(-\pi/3\) to \(5\pi/3\) adjusts the angle to correctly point in the fourth quadrant, reflecting the true position of the point.

Using the Arctan Function

The arctan function, or inverse tangent function, helps determine the angle formed by a specific ratio of opposite to adjacent sides in right-angled triangles. It is crucial when converting from rectangular to polar coordinates.

In our example, the point (2, -2√3) uses the formula: \(\theta = \arctan(y/x)\). Substituting the values: \(\theta = \arctan(-2\sqrt{3}/2)\) provides an angle of \(-\pi/3\) radians.

• This angle is initially negative, indicating direction in the opposite quadrant.
• To find the accurate angle in the fourth quadrant, add \(2\pi\), resulting in \(5\pi/3\).

This adjustment ensures the angle aligns with the correct quadrant, giving a true reflection of the point's polar coordinates.