Problem 50 - Precalculus with Limits

Question

In Exercises 37-54, a point in rectangular coordinates is given. Convert the point to polar coordinates. \(\left(-1, \sqrt{3}\right)\)

Step-by-Step Solution

Answer

The polar coordinates of the given point \((-1, \sqrt{3})\) are \((2, 2\pi/3)\).

Step 1: Calculate r

Firstly, calculate \(r\) which is the distance from the origin to the point. Apply the formula \(r = \sqrt{x^2 + y^2}\), where \(x = -1\) and \(y = \sqrt{3}\). Thus, it results in \(r = \sqrt{(-1)^2 + (\sqrt{3})^2} = 2\).

Step 2: Calculate Theta

Next, calculate \(\theta\) which is the angle measured from the positive x-axis to the point. Use the formula \(\theta = \arctan(\frac{y}{x})\), where \(y = \sqrt{3}\) and \(x=-1\). It is also important to note that as our point lies in the second quadrant, the \(\theta\) value calculated will be less than \(-\pi/2\), so we need to add \(\pi\) to get the actual angle in the second quadrant. Therefore, it becomes \(\theta = \arctan(-\sqrt{3}) + \pi = -\pi/3 + \pi = 2\pi/3\).

Step 3: Result in Polar Coordinates

Finally, the polar coordinates are given by \((r, \theta)\). So, substituting the obtained values, we get the polar coordinate as \((2, 2\pi/3)\).

Key Concepts

Polar Coordinates

Understanding the polar coordinate system is a fundamental part of mathematics, particularly when dealing with complex geometric problems. Polar coordinates represent points on a plane using a distance from a reference point (usually the origin) and an angle from a reference direction.

Imagine standing at the center of a circle and picking a point on its circumference. The radial distance from you (the center) to that point is the first component in polar coordinates, denoted as 'r'. The angle you turn from the positive x-axis (to your right, moving counterclockwise) to face that point directly is the second component, denoted as 'θ' or 'theta'. For a point in rectangular coordinates given as \(\left(x, y\right)\), we convert it to \(\left(r, \theta\right)\) in polar coordinates.

This approach provides a unique perspective in analyzing shapes and vectors, and is especially useful in fields such as physics, engineering, and computer graphics where rotational dynamics are more naturally described in polar terms.

Distance Formula

The distance formula is a foundational tool in geometry, allowing us to calculate the straight-line distance between two points in a plane. This formula, derived from the Pythagorean theorem, is expressed as \(r = \sqrt{x^2 + y^2}\) for points with rectangular coordinates \(\left(x, y\right)\).

When dealing with polar coordinates, the distance 'r' represents the radius of a circle on which a point lies and corresponds to how far that point is from the origin. To illustrate, if we have a point at \(\left(-1, \sqrt{3}\right)\), we would substitute these values into our formula, yielding \(r = \sqrt{(-1)^2 + (\sqrt{3})^2}\), simplifying to \(r = 2\). This constant radius is essential in polar graphs to determine the position of points relative to the center.

Angle Measurement in Coordinates

Angle measurement plays a critical role in the conversion from rectangular to polar coordinates. The angle in polar coordinates indicates the direction of the radial line from the origin to the point. To find this angle, denoted as \(\theta\), the formula \(\theta = \arctan(\frac{y}{x})\) is often used when converting a point from rectangular to polar coordinates.

However, this formula assumes the angle is referenced from the positive x-axis within the first quadrant. For points in other quadrants, adjustments are needed. As in the example of the point \(\left(-1, \sqrt{3}\right)\), which lies in the second quadrant, the initial calculation of \(\theta\) needs to have \(\pi\) added to represent the correct angle, since the arctan function only returns values from \(\frac{-\pi}{2}\) to \(\frac{\pi}{2}\). This adjustment results in \(\theta = -\frac{\pi}{3} + \pi = \frac{2\pi}{3}\), thus giving the precise angle measurement in polar coordinates.