Problem 27 - Precalculus with Calculus Previews

Question

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (1,-\sqrt{3}) $$

Step-by-Step Solution

Answer

The polar coordinates are: (a) \\(2, -\frac{\pi}{3}\\) (b) \\(-2, \frac{2\pi}{3}\\).

Step 1: Calculate the radius

To convert from rectangular coordinates \(x, y\) to polar coordinates \(r, \theta\), first calculate the radius \(r\) using the formula \(r = \sqrt{x^2 + y^2}\). Here, \(x = 1\) and \(y = -\sqrt{3}\), so, \\[ r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \].

Step 2: Calculate the angle with positive radius

Next, calculate the angle \(\theta\) using the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). For \(x = 1\) and \(y = -\sqrt{3}\), \\[ \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \]. \Since the angle is already in the range \(-\pi < \theta \leq \pi\) with \(r > 0\), the polar coordinates are \( (r, \theta) = (2, -\frac{\pi}{3}) \).

Step 3: Calculate the angle with negative radius

For \(r < 0\), we can use the relationship that \( (r, \theta) = (-r, \theta + \pi) \.\) The previously calculated \(r\) is \(-2\) and \(\theta\) is \(-\frac{\pi}{3}\), so add \pi\ to the angle: \\[ \theta + \pi = -\frac{\pi}{3} + \pi = \frac{2\pi}{3} \]. \Thus, the polar coordinates are \( (-r, \theta + \pi) = (-2, \frac{2\pi}{3}) \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to identify any point on a plane using two numbers. These two numbers are called the x-coordinate and the y-coordinate. They describe how far along and how far up or down the point is from the origin, which is the center of the plane where the x-axis and y-axis intersect.

Think of the x-axis as a horizontal line and the y-axis as a vertical line. A point like (1, \(-\sqrt{3}\)) tells you it's 1 unit to the right of the origin along the x-axis, and \(-\sqrt{3}\) units down along the y-axis. Notice that being down is negative, which is why we have a negative sign before \(\sqrt{3}\).

Understanding rectangular coordinates is essential as they are often a starting point for converting between different systems like polar coordinates. Remember: x is horizontal, y is vertical.

Radius Calculation

The radius in polar coordinates is called "r." It represents the distance from the origin to the point in question. Calculating the radius is crucial for converting rectangular coordinates to polar coordinates.

To find "r," we use the formula \(r = \sqrt{x^2 + y^2}\). This calculation gives us the length of the straight line connecting the origin to the point. This line is like the hypotenuse of a right triangle where "x" and "y" are the other two sides.

In our example, \(x = 1\) and \(y = -\sqrt{3}\). So, we calculate:

• \(1^2 = 1\)
• \((-\sqrt{3})^2 = 3\)
• Adding these values: \(1 + 3 = 4\)
• Taking the square root gives \(\sqrt{4} = 2\)

Thus, the radius is 2. Ensuring that we find the radius accurately allows for correct conversions.

Angle Calculation

Calculating the angle \(\theta\) helps us find out how far the point is rotated around the origin in polar coordinates. The angle is determined by the inverse tangent of \(\frac{y}{x}\), or \(\tan^{-1}\left(\frac{y}{x}\right)\). In simpler terms, it represents the direction of the line from the origin to the point, measured counterclockwise from the positive x-axis.

For our coordinates, \(x = 1\) and \(y = -\sqrt{3}\). So we find \(\theta\) using:

• \( \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3}) \)
• This results in \( \theta = -\frac{\pi}{3} \)

Since the reported angle \(-\frac{\pi}{3}\) is already within the defined range of \(-\pi < \theta \leq \pi\), no further adjustments are needed when \(r > 0\). For the case where \(r < 0\), we adjust by adding \(\pi\) to the angle to keep it aligned with the opposite direction.

Conversion Formulas

Converting between rectangular and polar coordinates involves two main formulas: one for the radius and one for the angle. Understanding these formulas is crucial.

• Radius Formula: \(r = \sqrt{x^2 + y^2}\) - measures the distance from the origin.
• Angle Formula: \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\) - finds the angle of rotation from the positive x-axis.

When dealing with polar coordinates, it's important to check which sector of the graph the point falls into. The formula for \(\theta\) gives the principal angle, but the resultant angle must comply with conditions such as \(-\pi < \theta \leq \pi\).

For a negative radius \(r < 0\), add \(\pi\) to \(\theta\) to maintain the correct direction of the components in the polar angle. This logical step is often required to ensure that the coordinates properly describe the position of a point. It's useful to practice using these formulas with different coordinate examples to become more familiar.