Problem 139 - Calculus Volume 3

Question

The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0, 2?]. Round to three decimal places. $$ (-6,8) $$

Step-by-Step Solution

Answer

First set: (10, 2.214). Second set: (10, 5.355).

Step 1: Understand the Relationship

Polar coordinates \(r, \theta\) can be related to rectangular coordinates \(x, y\) using the following relationships:- \ r = \sqrt{x^2 + y^2} \ - \ \theta = \arctan\left(\frac{y}{x}\right) \ We will use these to find two sets of polar coordinates.

Step 2: Calculate the Radius

The radius \(r\) is calculated as follows:\[ r = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

Step 3: Calculate the Angle

Use \(\theta = \arctan\left(\frac{y}{x}\right)\) to find the angular coordinate:\[ \theta = \arctan\left(\frac{8}{-6}\right) = \arctan\left(-\frac{4}{3}\right) \approx -0.927 \text{ radians} \]However, since the point \((-6, 8)\) is in the second quadrant, we add \ \pi \ to the angle to get a positive angle: \[ \theta = \pi - 0.927 \approx 2.214 \text{ radians} \]

Step 4: Adjust for the Range

Since the problem requests angles in \(0, 2\pi]\), the primary angle found \( \theta \approx 2.214 \) is already within this range. We need to find an additional set by adding \(2\pi\) if needed: For the second angle: First Set: \(r = 10, \theta = 2.214 \) \Second Set: Using \ (2\pi + \theta) \, we get \ (r = -10, \theta = 2.214 - \pi) = (10, 5.355) \

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, use a pair of numbers to define the position of a point in 2-dimensional space. The two numbers — usually denoted as \((x, y)\) — represent the horizontal and vertical distances of the point from the origin, which is the point \((0, 0)\) on the coordinate plane. In essence, these coordinates form right triangles with the x and y axes.
In the given exercise, the point is \((-6, 8)\). Here, the x-value is -6, indicating a point 6 units to the left of the y-axis, and y is 8, meaning the point is 8 units above the x-axis. This combination helps us in visualizing the point's exact position in the coordinate plane.
Understanding how rectangular coordinates work is essential since it allows us to transition to other coordinate systems, like polar coordinates, which provide different but equally important insights into spatial positions.

Angle Calculation

Angle calculation in the context of this problem involves finding the angle, also known as the angular coordinate, needed to convert between rectangular and polar coordinates. Once we have a point in rectangular form \((x, y)\), we can use trigonometric functions to find this angle, \(\theta\).
The basic formula for the angle is \(\theta = \arctan\left(\frac{y}{x}\right)\). Here, \(\theta\) represents the angle made with the positive x-axis. When calculating the angle for the coordinates \((-6, 8)\), \(\theta\) is found by substituting the given values:
\[\theta = \arctan\left(\frac{8}{-6}\right) = \arctan\left(-\frac{4}{3}\right) \approx -0.927 \text{ radians}\]
This calculation yields a negative angle, which hints towards the fourth quadrant. However, because \((-6, 8)\) is located in the second quadrant, we need to adjust the computed angle by adding \(\pi\) (180 degrees) to get \(\theta \approx 2.214 \) radians, properly reflecting its second quadrant position.

Quadrants in Trigonometry

In trigonometry, quadrants are important for determining the sign and value of angles. The coordinate plane is split into four quadrants, each defined by different combinations of positive and negative x and y values:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

Each trigonometric function behaves differently depending on which quadrant the angle resides in. For example, sine is positive in the first and second quadrants, while cosine is only positive in the first and fourth.
In the exercise, the point \((-6, 8)\) falls in the second quadrant, crucial information for finding the correct angle. Because the tangent is negative for coordinates \((-6, 8)\), without this quadrant consideration, one could mistakenly assume the point is in the fourth quadrant. Adjusting the angle using the correct quadrant ensures that the calculated polar coordinates accurately convey the point's position on the plane.