Problem 138 - 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. $$ (8,15) $$

Step-by-Step Solution

Answer

The polar coordinates for (8, 15) are (17, 1.081) and (17, 4.223).

Step 1: Understanding Rectangular and Polar Coordinates

Rectangular coordinates \((x, y)\) represent a point based on horizontal and vertical distances from the origin. Polar coordinates \( r, \theta \) use the distance from the origin and the angle from the positive x-axis.

Step 2: Calculate Radius using Pythagorean Theorem

The radius \(r\) in polar coordinates is the distance from the origin to the point. Calculate using \( r = \sqrt{x^2 + y^2} \). For \((8, 15)\), \( r = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \).

Step 3: Determine Initial Angle Using Arc Tangent Function

The angle \(\theta\) can be found by \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). For \((8, 15)\), \( \theta = \tan^{-1}\left(\frac{15}{8}\right) \). Compute \( \theta \approx 61.927^{\circ} \) or \( \theta \approx 1.081 \) radians.

Step 4: Adjust the Angle for Two Polar Coordinate Sets

In polar coordinates, angles can be expressed in multiple ways by adding or subtracting full rotations, i.e., \(2\pi\) radians. The first set is \((17, 1.081)\). The second set is \((17, 1.081 + 2\pi)\). Since \(1.081 + 2\pi > 2\pi\), another option is \((17, 1.081 - \pi)\) if applicable.

Step 5: Choose Angles Within the Interval \((0, 2\pi]\)

Ensure that angles you've obtained are within the interval \((0, 2\pi]\). Starting with \((17, 1.081)\), the first angle is already within range. For the second, since \(1.081 + \pi > 2\pi\), adjust to stay within bounds if you're choosing any other supplementary angle variation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two values: an x-coordinate and a y-coordinate. These two values represent the horizontal and vertical distances from the origin, which is the point (0,0). In the problem of converting coordinates, we're given the point (8, 15). This means:

• The x-coordinate is 8. This is the distance from the vertical y-axis, moving horizontally.
• The y-coordinate is 15. This indicates the distance from the horizontal x-axis, extending vertically.

Additionally, rectangular coordinates are generally used in problems involving algebra and geometry due to their straightforward representation of linear dimensions. They provide a simple way to pinpoint locations on a plane by moving along perpendicular axes.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially useful when dealing with right-angled triangles. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. More precisely, if a triangle has sides of length \(a\), \(b\), and hypotenuse \(c\), the theorem is:\[a^2 + b^2 = c^2\]In the context of converting rectangular coordinates into polar coordinates, the Pythagorean Theorem helps us find the radius \(r\), which represents the distance from the origin to the point in question. For example, given the rectangular coordinates (8, 15):

• We calculate \( r \) using \( r = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = 17 \).

Hence, \(r\) reflects the hypotenuse of the right triangle formed by these coordinates, providing a measure of distance in the polar system.

Arc Tangent Function

The arc tangent function, also known as the inverse tangent function, is essential for determining angles in trigonometry. It helps to find an angle whose tangent is the ratio of two given numbers. In a mathematical expression, this is represented as:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]For our point (8, 15), the angle \(\theta\) in radians is calculated by:

• \(\theta = \tan^{-1}\left(\frac{15}{8}\right)\)
• This results in an approximate angle of \(61.927^{\circ}\), or \(1.081\) radians.

The arc tangent function is pivotal in converting rectangular coordinates to polar coordinates because it translates the x and y measurements into a directional angle from the positive x-axis.

Radians

Radians are a unit of angular measure used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians measure angles as the length of the arc divided by the radius of the circle. One full circle corresponds to \(2\pi\) radians, making it a natural choice for measuring angles in trigonometry and calculus.In the given exercise, once we compute the angle \(\theta\) using the arc tangent function, we express it in radians for compatibility with the polar coordinate system. For the point (8, 15):

• \(\theta\) is approximately \(1.081\) radians.

When providing two sets of polar coordinates, one can be the initial angle \(\theta\), and the other can be adjusted by adding or subtracting \(2\pi\), as long as it remains within the interval \((0, 2\pi]\). Radians ensure that angles in mathematical equations are consistently represented, particularly in cyclic or periodic functions.