Problem 11 - Precalculus: Functions and Graphs

Question

Change the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta \leq 2 \pi\). (a) \((7,-7 \sqrt{3})\) (b) \((5,5)\)

Step-by-Step Solution

Answer

(a) Polar coordinates: (14, \(\frac{5\pi}{3}\)) (b) Polar coordinates: (5\sqrt{2}, \(\frac{\pi}{4}\))

Step 1: Understanding the Conversion

To convert from rectangular coordinates \((x,y)\) to polar coordinates \((r, \theta)\), we use the formulas:1. \(r = \sqrt{x^2 + y^2}\)2. \( \tan(\theta) = \frac{y}{x} \). After calculating \( \theta \), we must ensure it falls within the range \(0 \leq \theta \leq 2 \pi\). Let's apply these steps to the given points.

Step 2: Calculating r for (7, -7√3)

For the point \((7, -7\sqrt{3})\), calculate \(r\):\[ r = \sqrt{7^2 + (-7\sqrt{3})^2} = \sqrt{49 + 147} = \sqrt{196} = 14. \]

Step 3: Calculating θ for (7, -7√3)

Now, find \(\theta\) using \( \tan(\theta) = \frac{-7\sqrt{3}}{7} = -\sqrt{3}\).The angle that satisfies \(\tan(\theta) = -\sqrt{3}\) in the fourth quadrant (since \(y\) is negative) is \(\theta = \frac{5\pi}{3}\).

Step 4: Polar Coordinates for (7, -7√3)

Thus, the polar coordinates for \((7, -7\sqrt{3})\) are \((r, \theta) = (14, \frac{5\pi}{3})\).

Step 5: Calculating r for (5,5)

For the point \((5, 5)\), calculate \(r\):\[ r = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}. \]

Step 6: Calculating θ for (5,5)

Find \(\theta\) using \( \tan(\theta) = \frac{5}{5} = 1\).The angle that satisfies \(\tan(\theta) = 1\) in the first quadrant is \(\theta = \frac{\pi}{4}\).

Step 7: Polar Coordinates for (5,5)

Therefore, the polar coordinates for \((5, 5)\) are \((r, \theta) = (5\sqrt{2}, \frac{\pi}{4})\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent a point in a two-dimensional plane using two numbers. These two numbers typically labeled as \(x\) and \(y\), describe a point's location relative to two intersecting perpendicular lines known as axes. The \(x\)-axis is horizontal, and the \(y\)-axis is vertical.To understand this:

• Imagine a grid, like a piece of graph paper, where you can plot points using these coordinates.
• The first number (\(x\)) tells you how far to move horizontally (left or right) from the origin (the point where both axes intersect), and the second number (\(y\)) tells you how far to move vertically (up or down).

This system is incredibly intuitive and widely used in various fields such as mathematics, physics, and engineering. However, converting these coordinates into polar coordinates can be necessary for certain applications.

Angle Conversion

Converting an angle from a tangent ratio requires understanding how triangles and trigonometry work. When you are given a point in rectangular coordinates and need to find its polar equivalent, the angle \(\theta\) is obtained by using the tangent function:

• \( \tan(\theta) = \frac{y}{x} \)

This relationship is established because tangent in trigonometry is defined as opposite side over adjacent side in a right triangle.However, it's important to remember that calculators typically provide angles only in the range of \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This requires adjustments to find \(\theta\) in the range from \(0\) to \(2\pi\). Consider the quadrant the point is located in when you adjust the angle. Doing so ensures that angle conversion remains accurate and aligns with standard polar coordinate conventions.

Quadrant Identification

The Cartesian coordinate plane is divided into four quadrants. To identify which quadrant a point is in, examine the signs of the \(x\) and \(y\) coordinates:

• **Quadrant I:** Both \(x\) and \(y\) are positive.
• **Quadrant II:** \(x\) is negative, \(y\) is positive.
• **Quadrant III:** Both \(x\) and \(y\) are negative.
• **Quadrant IV:** \(x\) is positive, \(y\) is negative.

Understanding the quadrant is critical when determining the correct angle \(\theta\) for polar coordinates. Since angles in trigonometry are measured from the positive \(x\)-axis, identifying the correct quadrant ensures you account for the sign and position of the angle accurately.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. When converting rectangular coordinates to polar coordinates, use this theorem to find the magnitude \(r\) of the point from the origin:

• The formula is \( r = \sqrt{x^2 + y^2} \).

This formula comes from considering the \(x\) and \(y\) coordinates as the lengths of the two legs of a right triangle. The hypotenuse of this triangle, which is the straight-line distance from the origin to the point, represents \(r\).Applying the Pythagorean Theorem ensures you determine a positive and correct length \(r\), necessary when working with polar coordinates. Understanding this theorem can greatly enhance your problem-solving skills in not only coordinate conversion but also in other mathematical problems involving triangles.