Problem 29 - Technical Mathematics with Calculus

Question

Write the polar coordinates of each point. $$(-312,-509)$$

Step-by-Step Solution

Answer

(597.015, 4.16259)

Step 1: Determine the radius

Find the distance from the point to the origin using the Pythagorean theorem. The radius (r) can be calculated using the formula \( r = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the coordinates of the point. For the point (-312, -509), the radius would be \( r = \sqrt{(-312)^2 + (-509)^2} \).

Step 2: Calculate the radius

Calculate the radius using the values from step 1: \( r = \sqrt{(-312)^2 + (-509)^2} = \sqrt{97344 + 259081} = \sqrt{356425} \approx 597.015 \). Therefore, the radius is approximately 597.015.

Step 3: Determine the angle

The angle (\( \theta \)) in polar coordinates can be found using the arctangent function. Use the formula \( \theta = \arctan(\frac{y}{x}) \) for the point (-312, -509). Since both x and y are negative, the point lies in the third quadrant, and \( \theta \) should be adjusted by adding \( \pi \) radians to the result of the arctangent.

Step 4: Calculate the angle

Calculate the angle using the arctangent function: \( \theta = \arctan(\frac{-509}{-312}) \). Use a calculator to find \( \theta \approx \arctan(1.630769231) \approx 1.021 \) radians. To adjust for the third quadrant, add \( \pi \) radians: \( \theta = 1.021 + \pi \approx 1.021 + 3.14159 \approx 4.16259 \) radians.

Step 5: Write the polar coordinates

Combine the radius and angle to write the polar coordinates. The polar coordinates of the point (-312, -509) are \( (r, \theta) = (597.015, 4.16259) \).

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem provides a method to calculate the distance between two points in a plane, which is essential in converting Cartesian coordinates to polar coordinates.

When dealing with the conversion to polar coordinates, the Pythagorean theorem helps us find the radius, which is the distance from the origin to the point in question. Let's consider a point with Cartesian coordinates \( (x, y) \). The origin can be thought of as one corner of a right-angled triangle, with the point \( (x, y) \) forming the opposite corner, and the sides of the triangle being aligned with the axes. In this scenario, the radius \( r \) is the hypotenuse, and we can calculate it using the Pythagorean theorem, \( r = \sqrt{x^2 + y^2} \).

Radius Calculation

The process of calculating the radius is crucial in the conversion from Cartesian to polar coordinates. The radius, in this context, represents the direct line distance from the origin \( (0,0) \) to our point of interest in the coordinate plane. To compute this distance, we apply the Pythagorean theorem as previously mentioned.

For instance, if we have a point with coordinates \( (-312, -509) \), we determine the radius using the formula \( r = \sqrt{x^2 + y^2} \). We plug in the coordinates to obtain \( r = \sqrt{(-312)^2 + (-509)^2} \) and calculate it to find the exact value. The calculated radius not only informs us about the distance but also helps in plotting the point in polar coordinates, where every point is described by its distance from the origin (radius) and its angle relative to the positive x-axis (theta).

Arctangent Function

After finding the radius, we proceed to determine the angle using the arctangent function. The arctangent or inverse tangent function, denoted as \( \arctan \), is a trigonometric function that helps us find an angle whose tangent value is a given number. It's especially useful in polar coordinates since we often know the \( x \) and \( y \) values (from the Cartesian coordinates) but need to find the angle \( \theta \) they make with the x-axis.

The angle in polar coordinates is found using the formula \( \theta = \arctan(\frac{y}{x}) \). In our exercise, we calculate the angle for point \( (-312, -509) \) with \( \theta = \arctan(\frac{-509}{-312}) \). The negative signs indicate that the point is in the third quadrant. Therefore, we must adjust our angle by adding \( \pi \) radians to the initial arctangent result to correctly orient the angle within the coordinate system. This adjustment ensures that our final polar coordinates reflect the true position of the point in the plane.